diff --git a/2021-guangzhou.html b/2021-guangzhou.html
index 76d62a6..d2a37b8 100644
--- a/2021-guangzhou.html
+++ b/2021-guangzhou.html
@@ -159,7 +159,7 @@ <h1 class="Header__Title">2021年广州的风</h1>
   </header>
   <article id="https://www.notion.so/2fc609ec3d2c4529bd28ad01b8f11777" class="PageRoot"><div id="https://www.notion.so/97cd47386117430b87eb2b60ff9ba397" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F17e73390-46d6-45b2-88c1-5face1fbd613%2F4FEF92C1-70D0-4410-BDC2-F8C9B049875B.jpeg?width=4032&amp;table=block&amp;id=97cd4738-6117-430b-87eb-2b60ff9ba397"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F17e73390-46d6-45b2-88c1-5face1fbd613%2F4FEF92C1-70D0-4410-BDC2-F8C9B049875B.jpeg?width=4032&amp;table=block&amp;id=97cd4738-6117-430b-87eb-2b60ff9ba397" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/85c4afc4328a4cd1b859e8fba81ced8c" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">又在B站听到了“原来你也在这里”</span></span></p></div><div id="https://www.notion.so/9848dc0cc83745108c80eb4d63449218" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">那个夜晚，那些晚风和歌，成为了我一年后关在家里时最想回到的过去</span></span></p></div></article>
   <footer class="Footer">
-  <div>&copy; Patrick’s Blog 2023</div>
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diff --git a/about.html b/about.html
index e0fc3e9..a96622d 100644
--- a/about.html
+++ b/about.html
@@ -145,7 +145,7 @@ <h1 class="Header__Title">About me</h1>
   </header>
   <article id="https://www.notion.so/962db3a37ba549469b791b5bf14a1a9d" class="PageRoot"><h1 id="https://www.notion.so/61c2502cf1904549b7e7a9711614bcee" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--1"><a class="Anchor" href="#https://www.notion.so/61c2502cf1904549b7e7a9711614bcee"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">Hello! 👋</span></span></h1><div id="https://www.notion.so/6c9e033d8acc43d2936431b6ca339bad" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"></span></p></div><div id="https://www.notion.so/cb8e460009664c1289ebc6f668e26622" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">I am Heyang Gao, a second-year Ph.D. student at Renmin University of China, advised by Prof. Xu Chen. I received my Bachelor’s degree in Computer Science and Technology in Beijing University of Posts and Telecommunications. I am interested in causal discovery.</span></span></p></div><div id="https://www.notion.so/6b09bcc7deb9400c88d64dcc9d0b4b90" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"></span></p></div><div id="https://www.notion.so/625a1708284d4475bf1e523b8283e102" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F6522488a-5abe-4046-84b7-4ec6372ad85d%2FUntitled.jpeg?width=1276&amp;table=block&amp;id=625a1708-284d-4475-bf1e-523b8283e102"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F6522488a-5abe-4046-84b7-4ec6372ad85d%2FUntitled.jpeg?width=1276&amp;table=block&amp;id=625a1708-284d-4475-bf1e-523b8283e102" style="width:100%"/></a><figcaption><span class="SemanticStringArray"><span class="SemanticString">摸摸小鹿 @奈良若草山</span></span></figcaption></figure></div><div id="https://www.notion.so/aabd0cb8c03e4ab4a9078a19f28a4061" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"></span></p></div><div id="https://www.notion.so/0a49cb8ce626451fad39b71d35c0a578" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">My email: </span><span class="SemanticString"><a class="SemanticString__Fragment SemanticString__Fragment--Link" href="mailto:gaoheyang1014@gmail.com">gaoheyang1014@gmail.com</a></span></span></p></div><div id="https://www.notion.so/492c5dc3ed0c4586a496b04f920dcd9b" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"></span></p></div></article>
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diff --git a/adversarial-robustness-01.html b/adversarial-robustness-01.html
index 9b87993..2f0740e 100644
--- a/adversarial-robustness-01.html
+++ b/adversarial-robustness-01.html
@@ -271,7 +271,7 @@ <h1 class="Header__Title">[Adversarial Robustness] 1 Introduction to adversarial
 \equiv&amp;\min_\theta\frac{1}{|D_\mathrm{train}|}\sum_{(x,y)\in D_\mathrm{train}}\max_{\delta\in\Delta(x)}\ell(h_\theta(x+\delta),y).
 \end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.136323em;vertical-align:-2.3181615em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.8181615em;"><span style="top:-5.1928315000000005em;"><span class="pstrut" style="height:3.32144em;"></span><span class="mord"></span></span><span style="top:-2.8192835em;"><span class="pstrut" style="height:3.32144em;"></span><span class="mord"><span class="mrel">≡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.3181615em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.8181615em;"><span style="top:-5.1928315000000005em;"><span class="pstrut" style="height:3.32144em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.66786em;"><span style="top:-2.347892em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">min</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7521079999999999em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9467699999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span></span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">a</span><span class="mord mathrm mtight">d</span><span class="mord mathrm mtight" style="margin-right:0.01389em;">v</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31750199999999995em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">t</span><span class="mord mathrm mtight">r</span><span class="mord mathrm mtight">a</span><span class="mord mathrm mtight">i</span><span class="mord mathrm mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.8192835em;"><span class="pstrut" style="height:3.32144em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.66786em;"><span style="top:-2.347892em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">min</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7521079999999999em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31750199999999995em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">t</span><span class="mord mathrm mtight">r</span><span class="mord mathrm mtight">a</span><span class="mord mathrm mtight">i</span><span class="mord mathrm mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.808995em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">x</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span><span class="mclose mtight">)</span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3340428571428572em;"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathrm mtight">t</span><span class="mord mathrm mtight">r</span><span class="mord mathrm mtight">a</span><span class="mord mathrm mtight">i</span><span class="mord mathrm mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.516005em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.43055999999999994em;"><span style="top:-2.3089999999999997em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03785em;">δ</span><span class="mrel mtight">∈</span><span class="mord mtight">Δ</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">x</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">max</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.966em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">ℓ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord">.</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.3181615em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><div id="https://www.notion.so/3884ecb8ea5543b1971e90c866210792" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">我们将其称为对抗学习的</strong></span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">最小最大或鲁棒优化</em></strong></span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">形式化表述，在本教程的过程中我们会多次回到这个形式化表述。</strong></span></span></p></div><div id="https://www.notion.so/faed239a2b5b45679cf596bb7fce5b92" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">与传统训练一样，我们在实践中解决这个优化问题的方法是通过 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\theta"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span></span></span><span class="SemanticString"> 上的随机梯度下降。即，我们会重复地选择一个小批量 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="B\sube D_\mathrm{train}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>⊆</mo><msub><mi>D</mi><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">B\sube D_\mathrm{train}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.13597em;"></span><span class="mord mathdefault" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31750199999999995em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">t</span><span class="mord mathrm mtight">r</span><span class="mord mathrm mtight">a</span><span class="mord mathrm mtight">i</span><span class="mord mathrm mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">，并根据它的梯度更新 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\theta"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span></span></span></span></p></div><p id="https://www.notion.so/596faa316e2649ceb5fe5011aed38162" class="Equation" data-latex="\theta\coloneqq\theta-\frac{\alpha}{|B|}\sum_{(x,y)\in B}\nabla_\theta\max_{\delta\in\Delta(x)}\ell(h_\theta(x+\delta),y)."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi><mo><mi mathvariant="normal">≔</mi></mo><mi>θ</mi><mo>−</mo><mfrac><mi>α</mi><mrow><mi mathvariant="normal">∣</mi><mi>B</mi><mi mathvariant="normal">∣</mi></mrow></mfrac><munder><mo>∑</mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>B</mi></mrow></munder><msub><mi mathvariant="normal">∇</mi><mi>θ</mi></msub><munder><mo><mi>max</mi><mo>⁡</mo></mo><mrow><mi>δ</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></munder><mi mathvariant="normal">ℓ</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mi>θ</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>δ</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\theta\coloneqq\theta-\frac{\alpha}{|B|}\sum_{(x,y)\in B}\nabla_\theta\max_{\delta\in\Delta(x)}\ell(h_\theta(x+\delta),y).</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.03472em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.06666666666666667em;"></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.623565em;vertical-align:-1.516005em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.05017em;">B</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.808995em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">x</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span><span class="mclose mtight">)</span><span class="mrel mtight">∈</span><span class="mord mathdefault mtight" style="margin-right:0.05017em;">B</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.516005em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.43055999999999994em;"><span style="top:-2.3089999999999997em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03785em;">δ</span><span class="mrel mtight">∈</span><span class="mord mtight">Δ</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">x</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">max</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.966em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">ℓ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord">.</span></span></span></span></span></p><div id="https://www.notion.so/1a2ebedd40c043958a53983e1a8e232b" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">但是，考虑到内部函数自己包含着一个最大化问题，我们现在如何计算内部项的梯度呢？幸运的是，实践中答案非常简单，由 </span><span class="SemanticString"><a class="SemanticString__Fragment SemanticString__Fragment--Link" href="https://en.wikipedia.org/wiki/Danskin%27s_theorem">Danskin 定理</a></span><span class="SemanticString">给出。为了我们的讨论，这个定理指出，包含最大化项的内部函数的梯度简单地由在这个最大值处求值的函数的梯度给出。换句话说，令 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\delta^\star"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>δ</mi><mo>⋆</mo></msup></mrow><annotation encoding="application/x-tex">\delta^\star</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.688696em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">⋆</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 表示内部最优化问题的最优</span></span></p></div><p id="https://www.notion.so/b1f8b1b0e5fc4be79831a8e7c5152353" class="Equation" data-latex="\delta^\star=\argmax_{\delta\in\Delta(x)}\ell(h_\theta(x+\delta),y)"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>δ</mi><mo>⋆</mo></msup><mo>=</mo><munder><mo><mi mathvariant="normal">arg max</mi><mo>⁡</mo></mo><mrow><mi>δ</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></munder><mi mathvariant="normal">ℓ</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mi>θ</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>δ</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\delta^\star=\argmax_{\delta\in\Delta(x)}\ell(h_\theta(x+\delta),y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.738696em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.738696em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">⋆</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.91044em;vertical-align:-1.16044em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.43056000000000016em;"><span style="top:-2.11456em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03785em;">δ</span><span class="mrel mtight">∈</span><span class="mord mtight">Δ</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">x</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.0000000000000004em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop"><span class="mord mathrm">a</span><span class="mord mathrm">r</span><span class="mord mathrm" style="margin-right:0.01389em;">g</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathrm">m</span><span class="mord mathrm">a</span><span class="mord mathrm">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.16044em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">ℓ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span></span></p><div id="https://www.notion.so/02ddb44b4a394b0c987e5e548c6832dc" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">我们需要的梯度简单地由下式给出</span></span></p></div><p id="https://www.notion.so/0fc9a3ebea784f0b83de029d0cb2030b" class="Equation" data-latex="\nabla_\theta\max_{\delta\in\Delta(x)}\ell(h_\theta(x+\delta),y)=\nabla_\theta\ell(h_\theta(x+\delta^\star),y)"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">∇</mi><mi>θ</mi></msub><munder><mo><mi>max</mi><mo>⁡</mo></mo><mrow><mi>δ</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></munder><mi mathvariant="normal">ℓ</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mi>θ</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>δ</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi mathvariant="normal">∇</mi><mi>θ</mi></msub><mi mathvariant="normal">ℓ</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mi>θ</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><msup><mi>δ</mi><mo>⋆</mo></msup><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\nabla_\theta\max_{\delta\in\Delta(x)}\ell(h_\theta(x+\delta),y)=\nabla_\theta\ell(h_\theta(x+\delta^\star),y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.716em;vertical-align:-0.966em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.43055999999999994em;"><span style="top:-2.3089999999999997em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03785em;">δ</span><span class="mrel mtight">∈</span><span class="mord mtight">Δ</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">x</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">max</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.966em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">ℓ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">ℓ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.738696em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">⋆</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span></span></p><div id="https://www.notion.so/49032e85a7594a3aa5c736bda4fce577" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">（其中在右侧，我们把 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\delta^\star"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>δ</mi><mo>⋆</mo></msup></mrow><annotation encoding="application/x-tex">\delta^\star</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.688696em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">⋆</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 视作一个固定值，即我们不用担心它对 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\theta"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span></span></span><span class="SemanticString"> 的依赖）。这看起来可能是“显然的”，但实际上是很微妙的一点，要证明这一点成立并不容易（毕竟得到的 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\delta^\star"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>δ</mi><mo>⋆</mo></msup></mrow><annotation encoding="application/x-tex">\delta^\star</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.688696em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">⋆</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 的值依赖于 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\theta"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span></span></span><span class="SemanticString">，所以为什么在求梯度时可以把它视作独立于 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\theta"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span></span></span><span class="SemanticString"> 并不清晰）。我们在这里不会去证明 Danskin 定理，只是简单地指出这个性质会是我们的工作变得更简单。</span></span></p></div><div id="https://www.notion.so/c0d6d531eced4c2ba2ebd64d561875bb" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">给定这个框架，在寻找对抗样本和训练鲁棒的分类器的过程之间有一个很好的相互作用。具体来说，在经验对抗风险上梯度下降的过程看起来和下面的很像</span></span></p></div><ol class="NumberedListWrapper"><li id="https://www.notion.so/ad35ba0829dc416d9de28dbf47b70a52" class="NumberedList" value="1"><span class="SemanticStringArray"><span class="SemanticString">对于每个 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="x,y\in B"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">x,y\in B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335400000000001em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.05017em;">B</span></span></span></span></span></span><span class="SemanticString">，解内部最大化问题（即，计算一个对抗样本）</span></span><p id="https://www.notion.so/f593a4aa2ac94dd18a3f9169f5f2875f" class="Equation" data-latex="\delta^\star(x)=\argmax_{\delta\in\Delta(x)}\ell(h_\theta(x+\delta),y)."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>δ</mi><mo>⋆</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo><mi mathvariant="normal">arg max</mi><mo>⁡</mo></mo><mrow><mi>δ</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></munder><mi mathvariant="normal">ℓ</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mi>θ</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>δ</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\delta^\star(x)=\argmax_{\delta\in\Delta(x)}\ell(h_\theta(x+\delta),y).</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.738696em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">⋆</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.91044em;vertical-align:-1.16044em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.43056000000000016em;"><span style="top:-2.11456em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03785em;">δ</span><span class="mrel mtight">∈</span><span class="mord mtight">Δ</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">x</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.0000000000000004em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop"><span class="mord mathrm">a</span><span class="mord mathrm">r</span><span class="mord mathrm" style="margin-right:0.01389em;">g</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathrm">m</span><span class="mord mathrm">a</span><span class="mord mathrm">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.16044em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">ℓ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord">.</span></span></span></span></span></p></li><li id="https://www.notion.so/1f94c2b27a2e49e8859b6d3f24f32639" class="NumberedList" value="2"><span class="SemanticStringArray"><span class="SemanticString">计算经验对抗风险的梯度，并更新 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\theta"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span></span></span></span><p id="https://www.notion.so/266a04c4f2cb457eb31a70929e15012a" class="Equation" data-latex="\theta\coloneqq\theta-\frac{\alpha}{|B|}\sum_{(x,y)\in B}\nabla_\theta\ell(h_\theta(x+\delta^\star(x)),y)."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi><mo><mi mathvariant="normal">≔</mi></mo><mi>θ</mi><mo>−</mo><mfrac><mi>α</mi><mrow><mi mathvariant="normal">∣</mi><mi>B</mi><mi mathvariant="normal">∣</mi></mrow></mfrac><munder><mo>∑</mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>B</mi></mrow></munder><msub><mi mathvariant="normal">∇</mi><mi>θ</mi></msub><mi mathvariant="normal">ℓ</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mi>θ</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><msup><mi>δ</mi><mo>⋆</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\theta\coloneqq\theta-\frac{\alpha}{|B|}\sum_{(x,y)\in B}\nabla_\theta\ell(h_\theta(x+\delta^\star(x)),y).</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.03472em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.06666666666666667em;"></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.623565em;vertical-align:-1.516005em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.05017em;">B</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.808995em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">x</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span><span class="mclose mtight">)</span><span class="mrel mtight">∈</span><span class="mord mathdefault mtight" style="margin-right:0.05017em;">B</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.516005em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">ℓ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.738696em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">⋆</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord">.</span></span></span></span></span></p></li></ol><div id="https://www.notion.so/97b8a12a79f74816ab38138cd4f193a2" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">换句话说，我们反复地计算对抗样本，然后不仅基于原始数据点，而且基于这些对抗样本来更新分类器。这个过程已在深度学习文献中被称为“对抗训练”，并且（如果合适，稍后会详细介绍）这是我们用于训练对抗鲁棒模型的最有效的经验方法之一，尽管有一些注意事项值得一提。</span></span></p></div><div id="https://www.notion.so/3aa54330d1e3482bb752ce7da5a0d0aa" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">首先，我们应该注意到</span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">实际上</em></span><span class="SemanticString">，我们从来没有对真正的经验对抗风险进行梯度下降，正是因为我们通常不能最优地解决内部最大化问题。具体来说，如果像我们上面那样做通过梯度下降去做，内部最大化是一个非凸优化问题，当我们使用例如梯度下降这样的技术时，我们最多只能够找到局部最优。举例来说，由于只有当内部最大化问题被精确地解出时 Danskin 定理才在理论上使用，这似乎会给这样的方法带来问题。然而，在实践中，通常的情况是，如果内部优化问题解决得“足够好”，那么这个策略就会表现很好。不过，这在很大程度上取决于内部优化问题实际上的解决程度，如果只使用一个糟糕的近似策略来解决内部最大化问题，那么一个稍微更详尽的内部优化策略将被证明是一个很有效的攻击。这就是为什么当前最好的策略是尽可能明确地解出这个内部优化问题（甚至是近似地），使得后续的策略简单地“优化得更好”训练好的鲁棒性尽可能困难（尽管不是不可能）。</span></span></p></div><div id="https://www.notion.so/e6e0fd6a817a4a49b16a615d4bfd95eb" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">其次，虽然在理论上，我们可以将最坏情况的扰动作为计算梯度的点，但在实践中，这可能会导致训练过程的振荡，通常最好将多个具有不同随机初始化的扰动以及可能基于未扰动初始点的梯度结合起来使用。</span></span></p></div><div id="https://www.notion.so/d7a85da3012f4bb78638016aed06f3d7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">最后，我们应该注意到，一些鲁棒训练方法（特别是基于内部最大化问题上界的方法）实际上不需要反复寻找对抗样本点，然后再进行优化；相反，这些方法会产生一个内部最大化的闭式界限，可以通过非迭代方法解决。我们将在后续章节中更详细地讨论这些方法。</span></span></p></div><h3 id="https://www.notion.so/671d2ea2b1e64178aeb441922a86faeb" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/671d2ea2b1e64178aeb441922a86faeb"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">最后的注释</span></span></h3><div id="https://www.notion.so/74a2b2d313944a0ab1974009e1177d99" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">在继续之前，我们想再补充一下关于对抗鲁棒性的鲁棒优化形式化表述的价值的评论。重要的是要强调，</span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">每个对抗攻击和防御方法都分别是近似解决内部最大化和/或外部最小化问题的方法</strong></span><span class="SemanticString">。即使是没有明确表达这一点的论文，也在尝试解决这些问题（尽管可能存在一些潜在的差异，例如直接考虑不同的损失函数，如 0/1 损失而不是交叉熵损失）。</span></span></p></div><div id="https://www.notion.so/0c883ea6e9da4c3086444b778aa24555" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">在我们看来（这段话应该被理解为 Zico 和 Aleksander 的观点），该领域的一个显著挑战是许多论文从使用的</span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">方法</em></span><span class="SemanticString">的角度提出一种攻击或防御，而不是解决的</span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">问题</em></span><span class="SemanticString">（即优化问题）。这就是为什么我们会得到许多不同名称的策略，它们都考虑了上述优化的某些小变体，例如在 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\Delta(x)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Δ</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 项中考虑不同的范数界限，在解决内部最大化问题时使用不同的优化程序，或使用看似非常奢侈的技术来抵御攻击，这些技术通常似乎与优化形式化表述根本没有明显的关系。虽然有可能证明这种方法比我们已知的最佳策略更有效，但更为启发式的攻击和防御策略的历史并不乐观。</span></span></p></div><div id="https://www.notion.so/6381672a1b9044c98db80006ce4633c4" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">考虑到所有这些，本教程的下一章计划应该是清晰的。在第二章中，我们首先会稍微偏离主题，展示所有这些问题在</span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">线性</em></span><span class="SemanticString">模型的情况下是如何运作的；也许并不令人惊讶的是，在线性情况下，我们所讨论的内部最大化问题</span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">可以</em></span><span class="SemanticString">被准确地解决（或非常接近上界），我们可以对这些模型在对抗设定中的性能做出非常强有力的陈述。接下来，在第三章中，我们将回到深度网络的世界，研究内部最大化问题，重点关注可以应用的三类一般方法：1）下界（即构造对抗样本），2）精确解（通过组合优化），3）上界（通常使用一些更易处理的策略）。在第四章中，我们将解决训练对抗模型的问题，这通常涉及使用下界进行对抗训练，或者使用上界进行“证实的”鲁棒训练（使用精确组合解的对抗训练尚未被证明可行）。最后，在第五章中，我们回到本章中的一些更大的问题，并讨论更多内容：在这里，我们讨论对抗鲁棒性超越典型“安全”理由的价值；相反，我们考虑在正则化、泛化和学习到的表示的意义上，对抗鲁棒性的价值。</span></span></p></div></article>
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diff --git a/adversarial-robustness-02.html b/adversarial-robustness-02.html
index 2ee713f..67e5811 100644
--- a/adversarial-robustness-02.html
+++ b/adversarial-robustness-02.html
@@ -188,7 +188,7 @@ <h1 class="Header__Title">[Adversarial Robustness] 2 Linear models</h1>
 train_loader <span class="token operator">=</span> DataLoader<span class="token punctuation">(</span>mnist_train<span class="token punctuation">,</span> batch_size<span class="token operator">=</span><span class="token number">100</span><span class="token punctuation">,</span> shuffle<span class="token operator">=</span><span class="token boolean">True</span><span class="token punctuation">)</span>
 test_loader <span class="token operator">=</span> DataLoader<span class="token punctuation">(</span>mnist_test<span class="token punctuation">,</span> batch_size<span class="token operator">=</span><span class="token number">100</span><span class="token punctuation">,</span> shuffle<span class="token operator">=</span><span class="token boolean">False</span><span class="token punctuation">)</span></span></span></span></code></pre><div id="https://www.notion.so/2b12e2c933d541348b0207a2dfcafb25" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"></span></p></div></article>
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diff --git a/amoy.html b/amoy.html
index 0b6e3ad..72c0fda 100644
--- a/amoy.html
+++ b/amoy.html
@@ -159,7 +159,7 @@ <h1 class="Header__Title">厦门 Amoy ！</h1>
   </header>
   <article id="https://www.notion.so/bca694877d184da1a6c0744cfc858d4d" class="PageRoot"><div id="https://www.notion.so/d393d1a23ce943809d978b806bfb7b16" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F3bb198cf-f35f-4ba5-b684-580de70eda46%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=d393d1a2-3ce9-4380-9d97-8b806bfb7b16"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F3bb198cf-f35f-4ba5-b684-580de70eda46%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=d393d1a2-3ce9-4380-9d97-8b806bfb7b16" style="width:100%"/></a><figcaption><span class="SemanticStringArray"><span class="SemanticString">海边的小路 @黄厝沙滩</span></span></figcaption></figure></div><div id="https://www.notion.so/11eb2638f5864d568735e4d22d9ace4a" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F208c8c48-6ba9-4c84-b1b2-dd14528b3ce4%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=11eb2638-f586-4d56-8735-e4d22d9ace4a"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F208c8c48-6ba9-4c84-b1b2-dd14528b3ce4%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=11eb2638-f586-4d56-8735-e4d22d9ace4a" style="width:100%"/></a><figcaption><span class="SemanticStringArray"><span class="SemanticString">晚上的海，对面是小金门岛 @黄厝沙滩</span></span></figcaption></figure></div><div id="https://www.notion.so/22739bf40a6247e6a55e6f2a3b3efdf4" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fa704fcf4-6a66-490a-9ff9-e981f981ec4e%2FUntitled.jpeg?width=1707&amp;table=block&amp;id=22739bf4-0a62-47e6-a55e-6f2a3b3efdf4"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fa704fcf4-6a66-490a-9ff9-e981f981ec4e%2FUntitled.jpeg?width=1707&amp;table=block&amp;id=22739bf4-0a62-47e6-a55e-6f2a3b3efdf4" style="width:100%"/></a><figcaption><span class="SemanticStringArray"><span class="SemanticString">身后是厦门岛 @钟鼓索道</span></span></figcaption></figure></div><div id="https://www.notion.so/1225456ab7a04f5699c6ea220b806e4e" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fc5322207-33ef-40f9-a7be-43b17a237690%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=1225456a-b7a0-4f56-99c6-ea220b806e4e"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fc5322207-33ef-40f9-a7be-43b17a237690%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=1225456a-b7a0-4f56-99c6-ea220b806e4e" style="width:100%"/></a><figcaption><span class="SemanticStringArray"><span class="SemanticString">眺望厦门大学，羡慕！@厦门植物园山顶</span></span></figcaption></figure></div><div id="https://www.notion.so/2cfda9cf2bbe4490a5b7f267f56ca6d0" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Ff9c14360-ed24-4db2-b235-e396b0b4f020%2FUntitled.jpeg?width=1280&amp;table=block&amp;id=2cfda9cf-2bbe-4490-a5b7-f267f56ca6d0"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Ff9c14360-ed24-4db2-b235-e396b0b4f020%2FUntitled.jpeg?width=1280&amp;table=block&amp;id=2cfda9cf-2bbe-4490-a5b7-f267f56ca6d0" style="width:100%"/></a><figcaption><span class="SemanticStringArray"><span class="SemanticString">海上列车 @十里长堤</span></span></figcaption></figure></div><div id="https://www.notion.so/d6d8dbf0c7f5415c8051a40fbe6237d5" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fc20bf590-e84d-4eb9-af89-205a4def0b2a%2FUntitled.jpeg?width=1280&amp;table=block&amp;id=d6d8dbf0-c7f5-415c-8051-a40fbe6237d5"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fc20bf590-e84d-4eb9-af89-205a4def0b2a%2FUntitled.jpeg?width=1280&amp;table=block&amp;id=d6d8dbf0-c7f5-415c-8051-a40fbe6237d5" style="width:100%"/></a><figcaption><span class="SemanticStringArray"><span class="SemanticString">最美转角 @鼓浪屿</span></span></figcaption></figure></div><div id="https://www.notion.so/a9a793195f6643f48bb348b868712182" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F410047f6-0db9-4b8a-85f8-6d75f2b51fa3%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=a9a79319-5f66-43f4-8bb3-48b868712182"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F410047f6-0db9-4b8a-85f8-6d75f2b51fa3%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=a9a79319-5f66-43f4-8bb3-48b868712182" style="width:100%"/></a><figcaption><span class="SemanticStringArray"><span class="SemanticString">眺望厦门岛 @鼓浪屿日光岩</span></span></figcaption></figure></div><div id="https://www.notion.so/5806144fad7e400caabcf1ed7f0203c4" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fc59de625-f959-4bee-af56-f11c10a4c59c%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=5806144f-ad7e-400c-aabc-f1ed7f0203c4"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fc59de625-f959-4bee-af56-f11c10a4c59c%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=5806144f-ad7e-400c-aabc-f1ed7f0203c4" style="width:100%"/></a><figcaption><span class="SemanticStringArray"><span class="SemanticString">厦门 BRT 站牌 @第一码头站</span></span></figcaption></figure></div><div id="https://www.notion.so/1a2af0254c7c49f2877e5bebfe4cc9f3" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fc4cd678b-aa11-464a-a30e-ef5b2528cbf8%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=1a2af025-4c7c-49f2-877e-5bebfe4cc9f3"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fc4cd678b-aa11-464a-a30e-ef5b2528cbf8%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=1a2af025-4c7c-49f2-877e-5bebfe4cc9f3" style="width:100%"/></a><figcaption><span class="SemanticStringArray"><span class="SemanticString">完全没有人的沙滩 @观音山沙滩</span></span></figcaption></figure></div><div id="https://www.notion.so/04b779faf3aa4f66b29feed4157a061e" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F6ce4e682-fdb1-4577-b5a0-a19fc1043026%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=04b779fa-f3aa-4f66-b29f-eed4157a061e"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F6ce4e682-fdb1-4577-b5a0-a19fc1043026%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=04b779fa-f3aa-4f66-b29f-eed4157a061e" style="width:100%"/></a><figcaption><span class="SemanticStringArray"><span class="SemanticString">震惊，好美 @集美中学</span></span></figcaption></figure></div><div id="https://www.notion.so/59f8ed3a4d414a16ab485509decadb30" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F5bffdb06-d1dc-4e74-a41e-2abc61db8ad2%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=59f8ed3a-4d41-4a16-ab48-5509decadb30"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F5bffdb06-d1dc-4e74-a41e-2abc61db8ad2%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=59f8ed3a-4d41-4a16-ab48-5509decadb30" style="width:100%"/></a><figcaption><span class="SemanticStringArray"><span class="SemanticString">没吃到叶太 @集美大祠堂</span></span></figcaption></figure></div><div id="https://www.notion.so/c2c8e6072a9c464ba61fdb8f3516f694" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fb5a92d11-69b1-419b-bdf5-ed58a6c2ce48%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=c2c8e607-2a9c-464b-a61f-db8f3516f694"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fb5a92d11-69b1-419b-bdf5-ed58a6c2ce48%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=c2c8e607-2a9c-464b-a61f-db8f3516f694" style="width:100%"/></a><figcaption><span class="SemanticStringArray"><span class="SemanticString">二楼坐坐 @集美路边的一家咖啡厅</span></span></figcaption></figure></div><div id="https://www.notion.so/2a9cb540ee5840d48de2fdc7942e71c7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"></span></p></div></article>
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diff --git a/bayes.html b/bayes.html
index 84bea83..eb03f61 100644
--- a/bayes.html
+++ b/bayes.html
@@ -185,7 +185,7 @@ <h1 class="Header__Title">贝叶斯模型</h1>
 P(X_1=x_1,\dots, X_n=x_n) = \prod_{i=1}^n P(X_i=x_i|X_{i+1}=x_{i+1},\dots,X_n=x_n)"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>x</mi><mi>K</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>K</mi></msub><mi mathvariant="normal">∣</mi><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>x</mi><mrow><mi>K</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>…</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mi mathvariant="normal">∣</mi><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mspace linebreak="newline"></mspace><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub><mo>=</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mi>i</mi></msub><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mi mathvariant="normal">∣</mi><msub><mi>X</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub><mo>=</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(x_1,\dots,x_K)=p(x_K|x_1,\dots,x_{K-1})\dots p(x_2|x_1)p(x_1) \\
 P(X_1=x_1,\dots, X_n=x_n) = \prod_{i=1}^n P(X_i=x_i|X_{i+1}=x_{i+1},\dots,X_n=x_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em;">K</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em;">K</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em;">K</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.929066em;vertical-align:-1.277669em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000002em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∏</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.891661em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p></li></ul><div id="https://www.notion.so/1cfcf8e4047f48c7bede2fff6b540188" class="ColumnList"><div id="https://www.notion.so/08f6e9af99084ab884ea6e579c75452f" class="Column" style="width:calc((100% - var(--column-spacing) * 1) * 0.6875)"><ul class="BulletedListWrapper"><li id="https://www.notion.so/05ef25da5f274294a1266b6180d6883a" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">正常的贝叶斯网络：直观上有些边缺失</span></span><ul class="BulletedListWrapper"><li id="https://www.notion.so/4e18659c796548d5b522fa1277a48459" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="x_1"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">和</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="x_2"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">独立</span></span></li><li id="https://www.notion.so/8d464ddcd81946889f82a34ee82a6573" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="x_6"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>6</mn></msub></mrow><annotation encoding="application/x-tex">x_6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">和</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="x_7"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>7</mn></msub></mrow><annotation encoding="application/x-tex">x_7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">在</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="x_4"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>4</mn></msub></mrow><annotation encoding="application/x-tex">x_4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">给定的条件下独立</span></span></li><li id="https://www.notion.so/f3747674e7b444ac8f83a568138a9998" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="x_1,x_2,\dots,x_7"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>x</mi><mn>7</mn></msub></mrow><annotation encoding="application/x-tex">x_1,x_2,\dots,x_7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">的联合分布：</span></span><p id="https://www.notion.so/f96faa639d2447ba972eae7e4e726462" class="Equation" data-latex="p(x_1)p(x_2)p(x_3)p(x_4|x_1,x_2,x_3)p(x_5|x_1,x_3)p(x_6|x_4)p(x_7|x_4,x_5)"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>3</mn></msub><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>4</mn></msub><mi mathvariant="normal">∣</mi><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>3</mn></msub><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>5</mn></msub><mi mathvariant="normal">∣</mi><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>3</mn></msub><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>6</mn></msub><mi mathvariant="normal">∣</mi><msub><mi>x</mi><mn>4</mn></msub><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>7</mn></msub><mi mathvariant="normal">∣</mi><msub><mi>x</mi><mn>4</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>5</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(x_1)p(x_2)p(x_3)p(x_4|x_1,x_2,x_3)p(x_5|x_1,x_3)p(x_6|x_4)p(x_7|x_4,x_5)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p></li></ul></li></ul></div><div id="https://www.notion.so/19fe95f132e74e0a9fa14b6376bd2448" class="Column" style="width:calc((100% - var(--column-spacing) * 1) * 0.3125)"><div id="https://www.notion.so/512a94c98ba94be5a47db4569286be00" class="Image Image--Normal"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F90dd4167-b468-406f-8c45-e5b66c9649be%2FUntitled.png?width=354&amp;table=block&amp;id=512a94c9-8ba9-4be5-a47d-b4569286be00"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F90dd4167-b468-406f-8c45-e5b66c9649be%2FUntitled.png?width=354&amp;table=block&amp;id=512a94c9-8ba9-4be5-a47d-b4569286be00" style="width:354px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div></div></div><h3 id="https://www.notion.so/28e44ec55aa640e080a6fb3dce4c3985" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/28e44ec55aa640e080a6fb3dce4c3985"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">贝叶斯网络的结构形式</span></span></h3><div id="https://www.notion.so/78432b9784a84a7a8cf31da1b509c83b" class="ColumnList"><div id="https://www.notion.so/9625349dcc4b4bc19e4c5d3c00963867" class="Column" style="width:calc((100% - var(--column-spacing) * 1) * 0.75)"><ul class="BulletedListWrapper"><li id="https://www.notion.so/1da8b0314bf1438fb711c8cc0d38e9d8" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">head-to-head</span></span><div id="https://www.notion.so/202eb0fd04e640d08cf19cbb201c786d" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p(a,b,c)=p(a)p(b)p(c|a,b)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo separator="true">,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>c</mi><mi mathvariant="normal">∣</mi><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(a,b,c)=p(a)p(b)p(c|a,b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">c</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">b</span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">c</span><span class="mord">∣</span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">成立，即在</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="c"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">c</span></span></span></span></span></span><span class="SemanticString">未知的条件下，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a,b"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo separator="true">,</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a,b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span></span></span></span></span></span><span class="SemanticString">被阻断 (blocked) ，是独立的，称之为head-to-head条件独立。</span></span></p></div></li></ul></div><div id="https://www.notion.so/4f93b93c447247149ca8e88d35d8e80f" class="Column" style="width:calc((100% - var(--column-spacing) * 1) * 0.25)"><div id="https://www.notion.so/494e524a31254654bd40cc92d3fa632e" class="Image Image--Normal"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F59ce539c-fa39-49eb-af80-529531ad5a10%2FUntitled.png?width=144&amp;table=block&amp;id=494e524a-3125-4654-bd40-cc92d3fa632e"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F59ce539c-fa39-49eb-af80-529531ad5a10%2FUntitled.png?width=144&amp;table=block&amp;id=494e524a-3125-4654-bd40-cc92d3fa632e" style="width:144px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div></div></div><div id="https://www.notion.so/72b573a636a1440a8a4d37adf277e6a7" class="ColumnList"><div id="https://www.notion.so/6195e0f1a16c49e288b8e523ec4240fa" class="Column" style="width:calc((100% - var(--column-spacing) * 1) * 0.75)"><ul class="BulletedListWrapper"><li id="https://www.notion.so/541603398eb34cd28c5707e1cc6f1a53" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">tail-to-tail</span></span><ul class="BulletedListWrapper"><li id="https://www.notion.so/20f2899020364671a1a263a0382e3c3b" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">在</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="c"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">c</span></span></span></span></span></span><span class="SemanticString">未知的条件下：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p(a,b,c)=p(c)p(a|c)p(b|c)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo separator="true">,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mi mathvariant="normal">∣</mi><mi>c</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>b</mi><mi mathvariant="normal">∣</mi><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(a,b,c)=p(c)p(a|c)p(b|c)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">c</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">c</span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mord">∣</span><span class="mord mathdefault">c</span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">b</span><span class="mord">∣</span><span class="mord mathdefault">c</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">，此时，无法得出</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p(a,b)=p(a)p(b)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(a,b)=p(a)p(b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">b</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">，即</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="c"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">c</span></span></span></span></span></span><span class="SemanticString">未知时，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a,b"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo separator="true">,</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a,b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span></span></span></span></span></span><span class="SemanticString">不独立；</span></span></li><li id="https://www.notion.so/4d0ea1d657ec4dc8ad3cbb862a3007c1" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">在</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="c"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">c</span></span></span></span></span></span><span class="SemanticString">已知的条件下：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p(a,b|c)=\frac{p(a,b,c)}{p(c)}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mi mathvariant="normal">∣</mi><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo separator="true">,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">p(a,b|c)=\frac{p(a,b,c)}{p(c)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mord">∣</span><span class="mord mathdefault">c</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.53em;vertical-align:-0.52em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.01em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">c</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.485em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">a</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">b</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">c</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span><span class="SemanticString">，将</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p(a,b,c)=p(c)p(a|c)p(b|c)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo separator="true">,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mi mathvariant="normal">∣</mi><mi>c</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>b</mi><mi mathvariant="normal">∣</mi><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(a,b,c)=p(c)p(a|c)p(b|c)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">c</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">c</span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mord">∣</span><span class="mord mathdefault">c</span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">b</span><span class="mord">∣</span><span class="mord mathdefault">c</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">带入式子得到：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p(a,b|c)=p(a|c)p(b|c)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mi mathvariant="normal">∣</mi><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mi mathvariant="normal">∣</mi><mi>c</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>b</mi><mi mathvariant="normal">∣</mi><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(a,b|c)=p(a|c)p(b|c)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mord">∣</span><span class="mord mathdefault">c</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mord">∣</span><span class="mord mathdefault">c</span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">b</span><span class="mord">∣</span><span class="mord mathdefault">c</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">，即</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="c"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">c</span></span></span></span></span></span><span class="SemanticString">已知时，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a,b"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo separator="true">,</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a,b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span></span></span></span></span></span><span class="SemanticString">独立。</span></span></li></ul></li></ul></div><div id="https://www.notion.so/ecaa13e15d95426da2e7363891a38836" class="Column" style="width:calc((100% - var(--column-spacing) * 1) * 0.25000000000000006)"><div id="https://www.notion.so/6f9c7d0903fc43119e88d29bfd8b9680" class="Image Image--Normal"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F49d744e8-1759-4598-b6ee-89d8971971dc%2FUntitled.png?width=144&amp;table=block&amp;id=6f9c7d09-03fc-4311-9e88-d29bfd8b9680"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F49d744e8-1759-4598-b6ee-89d8971971dc%2FUntitled.png?width=144&amp;table=block&amp;id=6f9c7d09-03fc-4311-9e88-d29bfd8b9680" style="width:144px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div></div></div><div id="https://www.notion.so/a24a888bd12e4be6a07e8be86504dd4c" class="ColumnList"><div id="https://www.notion.so/56e6be29b8a547dd84ecd64ab0fb1549" class="Column" style="width:calc((100% - var(--column-spacing) * 1) * 0.6875)"><ul class="BulletedListWrapper"><li id="https://www.notion.so/c3f8654a53c1440387913246f548f4e2" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">head-to-tail</span></span><ul class="BulletedListWrapper"><li id="https://www.notion.so/3a262be9a0784ad98c0e2e1a01d0873a" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="c"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">c</span></span></span></span></span></span><span class="SemanticString">未知时：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p(a,b,c)=p(a)p(c|a)p(b|c)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo separator="true">,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>c</mi><mi mathvariant="normal">∣</mi><mi>a</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>b</mi><mi mathvariant="normal">∣</mi><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(a,b,c)=p(a)p(c|a)p(b|c)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">c</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">c</span><span class="mord">∣</span><span class="mord mathdefault">a</span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">b</span><span class="mord">∣</span><span class="mord mathdefault">c</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">，此时，无法得出</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p(a,b)=p(a)p(b)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(a,b)=p(a)p(b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">b</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">，即</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="c"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">c</span></span></span></span></span></span><span class="SemanticString">未知时，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a,b"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo separator="true">,</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a,b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span></span></span></span></span></span><span class="SemanticString">不独立；</span></span></li><li id="https://www.notion.so/dd43057e727d4b168e0df3ece225c8a8" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="c"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">c</span></span></span></span></span></span><span class="SemanticString">已知时：</span></span><div id="https://www.notion.so/0c06fbc94f154b9682faf1447ac07e9d" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p(a,b|c)=\frac{p(a,b,c)}{p(c)}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mi mathvariant="normal">∣</mi><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo separator="true">,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">p(a,b|c)=\frac{p(a,b,c)}{p(c)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mord">∣</span><span class="mord mathdefault">c</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.53em;vertical-align:-0.52em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.01em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">c</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.485em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">a</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">b</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">c</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span><span class="SemanticString">，且根据</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p(a,c)=p(a)p(c|a)=p(c)p(a|c)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>c</mi><mi mathvariant="normal">∣</mi><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mi mathvariant="normal">∣</mi><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(a,c)=p(a)p(c|a)=p(c)p(a|c)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">c</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">c</span><span class="mord">∣</span><span class="mord mathdefault">a</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">c</span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mord">∣</span><span class="mord mathdefault">c</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">，可得到：</span></span></p></div><div id="https://www.notion.so/e57b1a9ff41647ca9b28df491967a820" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p(a,b|c)=\frac{p(a,b,c)}{p(c)}=\frac{p(a)p(c|a)p(b|c)}{p(c)}=\frac{p(a,c)p(b|c)}{p(c)}=p(a|c)p(b|c)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mi mathvariant="normal">∣</mi><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo separator="true">,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>c</mi><mi mathvariant="normal">∣</mi><mi>a</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>b</mi><mi mathvariant="normal">∣</mi><mi>c</mi><mo stretchy="false">)</mo></mrow><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>b</mi><mi mathvariant="normal">∣</mi><mi>c</mi><mo stretchy="false">)</mo></mrow><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mi mathvariant="normal">∣</mi><mi>c</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>b</mi><mi mathvariant="normal">∣</mi><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(a,b|c)=\frac{p(a,b,c)}{p(c)}=\frac{p(a)p(c|a)p(b|c)}{p(c)}=\frac{p(a,c)p(b|c)}{p(c)}=p(a|c)p(b|c)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mord">∣</span><span class="mord mathdefault">c</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.53em;vertical-align:-0.52em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.01em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">c</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.485em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">a</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">b</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">c</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.53em;vertical-align:-0.52em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.01em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">c</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.485em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">a</span><span class="mclose mtight">)</span><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">c</span><span class="mord mtight">∣</span><span class="mord mathdefault mtight">a</span><span class="mclose mtight">)</span><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">b</span><span class="mord mtight">∣</span><span class="mord mathdefault mtight">c</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.53em;vertical-align:-0.52em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.01em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">c</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.485em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">a</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">c</span><span class="mclose mtight">)</span><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">b</span><span class="mord mtight">∣</span><span class="mord mathdefault mtight">c</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mord">∣</span><span class="mord mathdefault">c</span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">b</span><span class="mord">∣</span><span class="mord mathdefault">c</span><span class="mclose">)</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/704f68b01b3b4fe4a78fc242a2381be9" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">在</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="c"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">c</span></span></span></span></span></span><span class="SemanticString">给定的条件下，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a,b"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo separator="true">,</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a,b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span></span></span></span></span></span><span class="SemanticString">被阻断 (blocked) ，是独立的，称之为head-to-tail条件独立。</span></span></p></div></li></ul><div id="https://www.notion.so/f2eb6cf5c67d483280f53fb889edba10" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">head-to-tail是一个链式网络，在</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="x_i"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">给定的条件下，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="x_{i+1}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">x_{i+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">的分布和</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="x_1,x_2,\dots,x_{i-1}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>x</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">x_1,x_2,\dots,x_{i-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">条件独立。</span></span></p></div><div id="https://www.notion.so/d23c98623b9f4fcd8d1d8d14980d7b86" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="x_{i+1}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">x_{i+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">的分布状态只和</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="x_i"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">有关，和其他变量条件独立。</span></span></p></div><div id="https://www.notion.so/f6ec9754821145fd80efa75b87d9c379" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">当前状态只跟上一状态有关，跟上上或上上之前的状态无关——</span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">马尔科夫链</strong></span><span class="SemanticString"> (Markov chain)</span></span></p></div></li></ul></div><div id="https://www.notion.so/26c1bbe7f5f24d189b52ca4ca3e86fd8" class="Column" style="width:calc((100% - var(--column-spacing) * 1) * 0.31250000000000006)"><div id="https://www.notion.so/4ec9599568d84cd0bacdc1d9cbe24e0c" class="Image Image--Normal"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fad1df774-5249-4649-8173-b0f52507b6bd%2FUntitled.png?width=192&amp;table=block&amp;id=4ec95995-68d8-4cd0-bacd-c1d9cbe24e0c"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fad1df774-5249-4649-8173-b0f52507b6bd%2FUntitled.png?width=192&amp;table=block&amp;id=4ec95995-68d8-4cd0-bacd-c1d9cbe24e0c" style="width:192px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/aeabf815e450432789984ddcdb280de2" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"></span></p></div><div id="https://www.notion.so/cad63ba2ad4e480ea7081e5adceff71c" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"></span></p></div><div id="https://www.notion.so/da9e28cceff742a4961c8ec0dc6c6f59" class="Image Image--Normal"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fbaee29f5-7261-437b-be7a-3c48f61ad7ef%2FUntitled.png?width=336&amp;table=block&amp;id=da9e28cc-eff7-42a4-961c-8ec0dc6c6f59"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fbaee29f5-7261-437b-be7a-3c48f61ad7ef%2FUntitled.png?width=336&amp;table=block&amp;id=da9e28cc-eff7-42a4-961c-8ec0dc6c6f59" style="width:336px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/83e90941466343fca385fdfbf6391a0e" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"></span></p></div></div></div></article>
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   <article id="https://www.notion.so/ed3c5160cf624d62b561b8fb37aea5b0" class="PageRoot"><details id="https://www.notion.so/b0e1a523c57a4032b4b862e0acd76b36" class="ColorfulBlock ColorfulBlock--ColorDefault Toggle "><summary class="Toggle__Summary"><span class="SemanticStringArray"><span class="SemanticString">Machine Learning (4)</span></span></summary><div class="Toggle__Content"><div id="https://www.notion.so/b9d08f5e186542bdae534a420830bcac" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><a class="SemanticString__Fragment SemanticString__Fragment--Link" href="https://patrickg1014.github.io/decision-tree">🌲 决策树</a></span></span></p></div><div id="https://www.notion.so/6ea77861f6694d2aab82f52f47d29f8c" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><a class="SemanticString__Fragment SemanticString__Fragment--Link" href="https://patrickg1014.github.io/bayes">♠️ 贝叶斯模型</a></span></span></p></div><div id="https://www.notion.so/e2fa1d7f67de448c941d27578255210d" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><a class="SemanticString__Fragment SemanticString__Fragment--Link" href="https://patrickg1014.github.io/svm">🎽 支持向量机</a></span></span></p></div><div id="https://www.notion.so/b2933b2a43334c72919500c92f4fa26a" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><a class="SemanticString__Fragment SemanticString__Fragment--Link" href="https://patrickg1014.github.io/not-enough-data-01">🛸 数据不足下的学习 Part 1：半监督学习</a></span></span></p></div></div></details><details id="https://www.notion.so/b82a22ba310948b786d6f4e6cd8c001f" class="ColorfulBlock ColorfulBlock--ColorDefault Toggle "><summary 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diff --git a/causal.html b/causal.html
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--- a/causal.html
+++ b/causal.html
@@ -258,7 +258,7 @@ <h1 class="Header__Title">干预，在何处及如何进行？大规模非线性
 \boldsymbol{\mu}(x_{t+1})=\mathbf{k}_{t+1}^\top(\mathbf{K}+\mathbf{I}^{-1})\mathbf{U}_{\mathbf{x}^*}\\
 \boldsymbol{\sigma}^2(x_{t+1})=k(x_{t+1},x_{t+1})-\mathbf{k}_{t+1}^\top(\mathbf{K}+\mathbf{I})^{-1}\mathbf{k}_{t+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord text"><span class="mord">U</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.0593em;">G</span><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord text"><span class="mord">U</span></span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord mathbf">x</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.738696em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">U</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.28284em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathbf mtight">x</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6183428571428571em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1141079999999999em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.14736em;">N</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol">μ</span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">σ</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose">)</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">μ</span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.204439em;vertical-align:-0.305331em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">k</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.899108em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">⊤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">K</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.1141079999999999em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">I</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.864108em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord"><span class="mord mathbf">U</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.28284em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathbf mtight">x</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6183428571428571em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.1141079999999999em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">σ</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.204439em;vertical-align:-0.305331em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">k</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.899108em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">⊤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">K</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.1141079999999999em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathbf">I</span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.864108em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord mathbf">k</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></span></p><div id="https://www.notion.so/0f6036446f5a4eb89552168f916bd14b" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">对于高斯过程，我们使用了下面的超参数。Matern 核，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\text{length scale}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>length scale</mtext></mrow><annotation encoding="application/x-tex">\text{length scale}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">length scale</span></span></span></span></span></span></span><span class="SemanticString"> 1.0，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\text{length scale bounds}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>length scale bounds</mtext></mrow><annotation encoding="application/x-tex">\text{length scale bounds}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">length scale bounds</span></span></span></span></span></span></span><span class="SemanticString">（下=</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="1e-5"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi>e</mi><mo>−</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">1e-5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1</span><span class="mord mathdefault">e</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">5</span></span></span></span></span></span><span class="SemanticString">，上=</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="1e5"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi>e</mi><mn>5</mn></mrow><annotation encoding="application/x-tex">1e5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord mathdefault">e</span><span class="mord">5</span></span></span></span></span></span><span class="SemanticString">），</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\text{Nu}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Nu</mtext></mrow><annotation encoding="application/x-tex">\text{Nu}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord text"><span class="mord">Nu</span></span></span></span></span></span></span><span class="SemanticString">（学到的函数的平滑度）2.5。同时对核矩阵的对角线增加 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="1e-6"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi>e</mi><mo>−</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">1e-6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1</span><span class="mord mathdefault">e</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">6</span></span></span></span></span></span><span class="SemanticString">。</span></span></p></div><div id="https://www.notion.so/63f3e7dc6c374452a245dcb95346c4a6" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"></span></p></div></article>
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diff --git a/changsha.html b/changsha.html
index e353332..220ed9d 100644
--- a/changsha.html
+++ b/changsha.html
@@ -159,7 +159,7 @@ <h1 class="Header__Title">长沙吃吃逛逛</h1>
   </header>
   <article id="https://www.notion.so/c2bba64f462d4564910fe8023a0a2f91" class="PageRoot"><div id="https://www.notion.so/4bed36ee20eb46cba0ecfc382214ba76" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">决定把去长沙的过程写得详细一点，因为很喜欢长沙，特别是住在河西金融中心人少的地方😂</span></span></p></div><div id="https://www.notion.so/66acfd60d8454dfe93e5912c1a0d327d" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">第一天晚上从大连飞去长沙，在黄花机场第一次坐了磁浮列车，速度和乘坐体验都和北京首都机场线差不多🤣。酒店订在了 4 号线茶子山地铁站附近的全季酒店，旁边是凯德壹中心，里面的茶颜悦色几乎不用排队，很不错😋，柠季的鸭屎香柠檬茶也很好喝。重要的是超级喜欢益阳麻辣烫，特别是加在里面的粉，外卖随便点的臭豆腐也好吃。 </span></span></p></div><div id="https://www.notion.so/8e514885539f45d1a50c178ad5a97e16" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">第二天早上吃了金融中心附近很贵的炒码面，味道还可以。上午去了湖南博物院，快速参观了湖南历史和辛追夫人展。</span></span></p></div><div id="https://www.notion.so/13220d32582143ec9c5254a75c903223" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fd3d0c8c6-df04-4d57-9dde-1d7a4ebef790%2FUntitled.jpeg?width=672&amp;table=block&amp;id=13220d32-5821-43ec-9c52-54a75c903223"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fd3d0c8c6-df04-4d57-9dde-1d7a4ebef790%2FUntitled.jpeg?width=672&amp;table=block&amp;id=13220d32-5821-43ec-9c52-54a75c903223" style="width:100%"/></a><figcaption><span class="SemanticStringArray"><span class="SemanticString">女书 @湖南博物院</span></span></figcaption></figure></div><div id="https://www.notion.so/27f6856e78854ab78828a65f6d7c1a30" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">看了王师傅的视频第一次去吃豪客来牛排，在辽宁和北京都没见过hhh👀 好开心</span></span></p></div><div id="https://www.notion.so/99a8077dec094537a8caf4baa7b954bf" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fbb3270e5-511b-4269-a8de-be86136bebd9%2FUntitled.jpeg?width=768&amp;table=block&amp;id=99a8077d-ec09-4537-a8ca-f4baa7b954bf"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fbb3270e5-511b-4269-a8de-be86136bebd9%2FUntitled.jpeg?width=768&amp;table=block&amp;id=99a8077d-ec09-4537-a8ca-f4baa7b954bf" style="width:100%"/></a><figcaption><span class="SemanticStringArray"><span class="SemanticString">很像乡里别 @豪客来牛排</span></span></figcaption></figure></div><div id="https://www.notion.so/869ff25867c047f089e7ac4811e4ad4c" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">晚上去长郡中学看了一下，就在黄兴广场附近，羡慕长郡的高中生买小吃很方便。</span></span></p></div><div id="https://www.notion.so/7d29f51af09746bc8a72ee3da0fc95e6" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fb271d98b-8d53-4ec3-8518-3e1bac8cda5a%2FUntitled.jpeg?width=4032&amp;table=block&amp;id=7d29f51a-f097-46bc-8a72-ee3da0fc95e6"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fb271d98b-8d53-4ec3-8518-3e1bac8cda5a%2FUntitled.jpeg?width=4032&amp;table=block&amp;id=7d29f51a-f097-46bc-8a72-ee3da0fc95e6" style="width:100%"/></a><figcaption><span class="SemanticStringArray"><span class="SemanticString">冒充长郡高中生 @长郡中学</span></span></figcaption></figure></div><div id="https://www.notion.so/d3cc367d1e4c412da28e374b92a1bbff" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">还顺路去了很多地方啦，比如步行街、南门口、杜甫江阁，又沿着江走了走，从橘子洲大桥上走到橘子洲的hhh，并没有坐地铁去。橘子洲晚上去也是很多人，都是来打卡合影的。</span></span></p></div><div id="https://www.notion.so/aa7c34bde75f4a12b9153c773156d24a" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F2f88028c-43dd-4e78-8a49-ead44bc2f64a%2FUntitled.jpeg?width=4032&amp;table=block&amp;id=aa7c34bd-e75f-4a12-b915-3c773156d24a"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F2f88028c-43dd-4e78-8a49-ead44bc2f64a%2FUntitled.jpeg?width=4032&amp;table=block&amp;id=aa7c34bd-e75f-4a12-b915-3c773156d24a" style="width:100%"/></a><figcaption><span class="SemanticStringArray"><span class="SemanticString">傻傻的游客照 @橘子洲头</span></span></figcaption></figure></div><div id="https://www.notion.so/8169c4e0166c4d2484d3a2b142ba9299" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">后面几天又吃了炊烟，感觉确实比在北京吃的费大厨好吃一点hhh，可能因为我比较爱吃牛肉一点。在 IFS 逛了逛，在超级文和友逛了逛，在湖南大学逛了逛，在岳麓山狼狈地爬了爬（下雨天跟着高德地图爬了一条泥巴路，真的很蠢）。</span></span></p></div><div id="https://www.notion.so/a4f2733c9d204083a0ab930b2521317e" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">中间抽出一天时间去了张家界，从观沙岭坐城铁就可以到，感觉湖南的城际铁路修的真不错👍</span></span></p></div><div id="https://www.notion.so/dedd11e9116c41d180127ac6977c4f03" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">去了天门山景区，结果山上又是下雨，雾蒙蒙的，不过人很少，我觉得也很有趣hhhh</span></span></p></div><div id="https://www.notion.so/6ee52b442b11458994cba8f7236e16a3" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fda5b8608-b7ce-4c5d-a0fb-2e1e539b923d%2FUntitled.jpeg?width=4032&amp;table=block&amp;id=6ee52b44-2b11-4589-94cb-a8f7236e16a3"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fda5b8608-b7ce-4c5d-a0fb-2e1e539b923d%2FUntitled.jpeg?width=4032&amp;table=block&amp;id=6ee52b44-2b11-4589-94cb-a8f7236e16a3" style="width:100%"/></a><figcaption><span class="SemanticStringArray"><span class="SemanticString">工作人员教的游客照手势 @天门山</span></span></figcaption></figure></div><div id="https://www.notion.so/633727a50dbc4f8589395d2318ff0d83" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">金色印象捏捏全身，抄手、酸辣粉都巨好吃，一定要再去吃！😋</span></span></p></div><div id="https://www.notion.so/2c2854e45e1a42ee9a20d9600473e656" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">最后想说大连太美啦，特别是从飞机上看，真的好喜欢大连🥰</span></span></p></div><div id="https://www.notion.so/bf5af7868cb346588549157dc7d0c1c0" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F3414d395-37d1-40d6-8f36-3b54173499af%2FUntitled.jpeg?width=4032&amp;table=block&amp;id=bf5af786-8cb3-4658-8549-157dc7d0c1c0"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F3414d395-37d1-40d6-8f36-3b54173499af%2FUntitled.jpeg?width=4032&amp;table=block&amp;id=bf5af786-8cb3-4658-8549-157dc7d0c1c0" style="width:100%"/></a><figcaption><span class="SemanticStringArray"><span class="SemanticString">大连 @飞机</span></span></figcaption></figure></div><div id="https://www.notion.so/be152702e7a946b5a8bcb492eb0ee24b" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"></span></p></div></article>
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diff --git a/dad-rl-01.html b/dad-rl-01.html
index 618aefa..125046b 100644
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+++ b/dad-rl-01.html
@@ -174,7 +174,7 @@ <h1 class="Header__Title">DAD 与贝叶斯强化学习</h1>
  151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.01100000000000001em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">，智能体收到一个随机奖励 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="r_t\sim R(r|s_t,a_t,s_{t+1})\in\mathbb{R}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mi>t</mi></msub><mo>∼</mo><mi>R</mi><mo stretchy="false">(</mo><mi>r</mi><mi mathvariant="normal">∣</mi><msub><mi>s</mi><mi>t</mi></msub><mo separator="true">,</mo><msub><mi>a</mi><mi>t</mi></msub><mo separator="true">,</mo><msub><mi>s</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow><annotation encoding="application/x-tex">r_t\sim R(r|s_t,a_t,s_{t+1})\in\mathbb{R}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68889em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbb">R</span></span></span></span></span></span></span><span class="SemanticString">。智能体的目标是要最大化奖励的折扣和 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\sum_{t=0}^T\gamma^tr_t"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow><mi>T</mi></msubsup><msup><mi>γ</mi><mi>t</mi></msup><msub><mi>r</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\sum_{t=0}^T\gamma^tr_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2809409999999999em;vertical-align:-0.29971000000000003em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">。考虑到问题的马尔可夫性，根据某个</span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">策略</em></span><span class="SemanticString"> </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\pi"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span></span></span></span></span></span><span class="SemanticString"> 选择行动就足够了，其中 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a_t=\pi(s_t)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>t</mi></msub><mo>=</mo><mi>π</mi><mo stretchy="false">(</mo><msub><mi>s</mi><mi>t</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a_t=\pi(s_t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">。对于一个策略的最优条件是 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\pi^*=\argmax_\pi\mathcal{J}(\pi)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>π</mi><mo>∗</mo></msup><mo>=</mo><msub><mo><mi mathvariant="normal">arg max</mi><mo>⁡</mo></mo><mi>π</mi></msub><mi mathvariant="script">J</mi><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi^*=\argmax_\pi\mathcal{J}(\pi)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.688696em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.688696em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop"><span class="mop"><span class="mord mathrm">a</span><span class="mord mathrm">r</span><span class="mord mathrm" style="margin-right:0.01389em;">g</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathrm">m</span><span class="mord mathrm">a</span><span class="mord mathrm">x</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.057252em;"><span style="top:-2.4558600000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">π</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.18472em;">J</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">，其中</span></span></p></div><p id="https://www.notion.so/9d3b19c469424a7cb1d64697b438ee4b" class="Equation" data-latex="\mathcal{J}(\pi)=\mathbb{E}_{s_0\sim p(s_0)\prod_{t=0}^Ta_t=\pi(s_t),s_{t+1}\sim\mathcal{P}(s|s_t,a_t),r_t\sim R(r|s_t,a_t,s_{t+1})}\left[\sum_{t=0}^T\gamma^tr_t\right]."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">J</mi><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi mathvariant="double-struck">E</mi><mrow><msub><mi>s</mi><mn>0</mn></msub><mo>∼</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>s</mi><mn>0</mn></msub><mo stretchy="false">)</mo><munderover><mo>∏</mo><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow><mi>T</mi></munderover><msub><mi>a</mi><mi>t</mi></msub><mo>=</mo><mi>π</mi><mo stretchy="false">(</mo><msub><mi>s</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><msub><mi>s</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>∼</mo><mi mathvariant="script">P</mi><mo stretchy="false">(</mo><mi>s</mi><mi mathvariant="normal">∣</mi><msub><mi>s</mi><mi>t</mi></msub><mo separator="true">,</mo><msub><mi>a</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><msub><mi>r</mi><mi>t</mi></msub><mo>∼</mo><mi>R</mi><mo stretchy="false">(</mo><mi>r</mi><mi mathvariant="normal">∣</mi><msub><mi>s</mi><mi>t</mi></msub><mo separator="true">,</mo><msub><mi>a</mi><mi>t</mi></msub><mo separator="true">,</mo><msub><mi>s</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msub><mrow><mo fence="true">[</mo><munderover><mo>∑</mo><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow><mi>T</mi></munderover><msup><mi>γ</mi><mi>t</mi></msup><msub><mi>r</mi><mi>t</mi></msub><mo fence="true">]</mo></mrow><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\mathcal{J}(\pi)=\mathbb{E}_{s_0\sim p(s_0)\prod_{t=0}^Ta_t=\pi(s_t),s_{t+1}\sim\mathcal{P}(s|s_t,a_t),r_t\sim R(r|s_t,a_t,s_{t+1})}\left[\sum_{t=0}^T\gamma^tr_t\right].</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.18472em;">J</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mord"><span class="mord"><span class="mord mathbb">E</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.34480000000000005em;"><span style="top:-2.4251350000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mrel mtight">∼</span><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mclose mtight">)</span><span class="mspace mtight" style="margin-right:0.19516666666666668em;"></span><span class="mop mtight"><span class="mop op-symbol small-op mtight" style="position:relative;top:-0.0000050000000000050004em;">∏</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8852357142857143em;"><span style="top:-2.1785614285714283em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-2.8971428571428572em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.32143857142857146em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.19516666666666668em;"></span><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.29634285714285713em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mrel mtight">=</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">π</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.29634285714285713em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mclose mtight">)</span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173142857142857em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.20252142857142857em;"><span></span></span></span></span></span></span><span class="mrel mtight">∼</span><span class="mord mtight"><span class="mord mathcal mtight" style="margin-right:0.08222em;">P</span></span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">s</span><span class="mord mtight">∣</span><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.29634285714285713em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.29634285714285713em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mclose mtight">)</span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.29634285714285713em;"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mrel mtight">∼</span><span class="mord mathdefault mtight" style="margin-right:0.00773em;">R</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mtight">∣</span><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.29634285714285713em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.29634285714285713em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173142857142857em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.20252142857142857em;"><span></span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.49987200000000004em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">[</span></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">]</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">.</span></span></span></span></span></p><div id="https://www.notion.so/cb0c964ebb9948cbb1420777cf61788c" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">在经典 MDP 中，我们假定 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathcal{P}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">P</mi></mrow><annotation encoding="application/x-tex">\mathcal{P}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span></span></span></span></span></span><span class="SemanticString"> 和 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="R"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span></span></span></span></span></span><span class="SemanticString"> 在规划阶段是已知的，当智能体设计它们的策略 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\pi"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span></span></span></span></span></span><span class="SemanticString"> 时。在规划策略时特别有用的是价值函数，定义为</span></span></p></div><p id="https://www.notion.so/bbce95a7b28446f1a06b941dfcca3d32" class="Equation" data-latex="V^\pi(s)=\mathbb{E}_{s&#x27;\sim\mathcal{P}(\cdot|s,\pi(s)),r\sim R(r|s,\pi(s),s&#x27;)}[r+\gamma V^\pi(s&#x27;)]"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>V</mi><mi>π</mi></msup><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi mathvariant="double-struck">E</mi><mrow><msup><mi>s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>∼</mo><mi mathvariant="script">P</mi><mo stretchy="false">(</mo><mo>⋅</mo><mi mathvariant="normal">∣</mi><mi>s</mi><mo separator="true">,</mo><mi>π</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>r</mi><mo>∼</mo><mi>R</mi><mo stretchy="false">(</mo><mi>r</mi><mi mathvariant="normal">∣</mi><mi>s</mi><mo separator="true">,</mo><mi>π</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><msup><mi>s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">[</mo><mi>r</mi><mo>+</mo><mi>γ</mi><msup><mi>V</mi><mi>π</mi></msup><mo stretchy="false">(</mo><msup><mi>s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">V^\pi(s)=\mathbb{E}_{s&#x27;\sim\mathcal{P}(\cdot|s,\pi(s)),r\sim R(r|s,\pi(s),s&#x27;)}[r+\gamma V^\pi(s&#x27;)]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">π</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">s</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1052em;vertical-align:-0.3551999999999999em;"></span><span class="mord"><span class="mord"><span class="mord mathbb">E</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.34480000000000005em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mrel mtight">∼</span><span class="mord mtight"><span class="mord mathcal mtight" style="margin-right:0.08222em;">P</span></span><span class="mopen mtight">(</span><span class="mord mtight">⋅</span><span class="mord mtight">∣</span><span class="mord mathdefault mtight">s</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">π</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">s</span><span class="mclose mtight">)</span><span class="mclose mtight">)</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mrel mtight">∼</span><span class="mord mathdefault mtight" style="margin-right:0.00773em;">R</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mtight">∣</span><span class="mord mathdefault mtight">s</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">π</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">s</span><span class="mclose mtight">)</span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3551999999999999em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.051892em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.05556em;">γ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">π</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.801892em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose">]</span></span></span></span></span></p><div id="https://www.notion.so/b2853fcb70db4a7da63a5f91ffc94c35" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">和 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="Q"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">Q</span></span></span></span></span></span><span class="SemanticString"> 函数</span></span></p></div><p id="https://www.notion.so/9f18a877bbfc41e2ad97699cda052bcc" class="Equation" data-latex="Q^\pi(s,a)=\mathbb{E}_{s&#x27;\sim\mathcal{P}(\cdot|s,a),r\sim R(r|s,a,s&#x27;)}[r+\gamma V^\pi(s&#x27;)]."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>Q</mi><mi>π</mi></msup><mo stretchy="false">(</mo><mi>s</mi><mo separator="true">,</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi mathvariant="double-struck">E</mi><mrow><msup><mi>s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>∼</mo><mi mathvariant="script">P</mi><mo stretchy="false">(</mo><mo>⋅</mo><mi mathvariant="normal">∣</mi><mi>s</mi><mo separator="true">,</mo><mi>a</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>r</mi><mo>∼</mo><mi>R</mi><mo stretchy="false">(</mo><mi>r</mi><mi mathvariant="normal">∣</mi><mi>s</mi><mo separator="true">,</mo><mi>a</mi><mo separator="true">,</mo><msup><mi>s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">[</mo><mi>r</mi><mo>+</mo><mi>γ</mi><msup><mi>V</mi><mi>π</mi></msup><mo stretchy="false">(</mo><msup><mi>s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">Q^\pi(s,a)=\mathbb{E}_{s&#x27;\sim\mathcal{P}(\cdot|s,a),r\sim R(r|s,a,s&#x27;)}[r+\gamma V^\pi(s&#x27;)].</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">π</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">a</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1052em;vertical-align:-0.3551999999999999em;"></span><span class="mord"><span class="mord"><span class="mord mathbb">E</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.34480000000000005em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mrel mtight">∼</span><span class="mord mtight"><span class="mord mathcal mtight" style="margin-right:0.08222em;">P</span></span><span class="mopen mtight">(</span><span class="mord mtight">⋅</span><span class="mord mtight">∣</span><span class="mord mathdefault mtight">s</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">a</span><span class="mclose mtight">)</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mrel mtight">∼</span><span class="mord mathdefault mtight" style="margin-right:0.00773em;">R</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mtight">∣</span><span class="mord mathdefault mtight">s</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">a</span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3551999999999999em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.051892em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.05556em;">γ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">π</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.801892em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose">]</span><span class="mord">.</span></span></span></span></span></p><div id="https://www.notion.so/6a5e0c351e7942b287dcda4aea109b52" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">当 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="T=\infin"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi><mo>=</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">T=\infin</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord">∞</span></span></span></span></span></span><span class="SemanticString"> 时这些公式是有效的，对于有限时间视野我们还要在状态评估中考虑时间 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="t"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span></span></span></span></span></span><span class="SemanticString">。</span></span></p></div><h3 id="https://www.notion.so/65d8d37c6e5040f7868f730eb89436c4" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/65d8d37c6e5040f7868f730eb89436c4"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">贝叶斯适应性马尔可夫决策过程</span></span></h3><div id="https://www.notion.so/ab3d4c19bb2949f7bcd32d82e2e7df62" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><a class="SemanticString__Fragment SemanticString__Fragment--Link" href="https://scholarworks.umass.edu/dissertations/AAI3039353/">BAMDP</a></span><span class="SemanticString"> 是一种将 MDP 泛化以解决未知转移模型的方法。在 BAMDP 中，智能体保存一个转移模型上的显式的后验分布，称为信念状态。这允许一种形式上优雅的方法来处理不确定性下的行为，这种方法可以权衡探索（学习转移模型）和利用（执行获得高奖励的行为）。</span></span></p></div><div id="https://www.notion.so/7fe213385a7847bb958886bccd5d1a82" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">为了使用 </span><span class="SemanticString"><a class="SemanticString__Fragment SemanticString__Fragment--Link" href="https://arxiv.org/abs/1205.3109">Guez 等人</a></span><span class="SemanticString">的符号正式地建立起这个概念，我们首先考虑一个更外一层的在先验为 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="P(\mathcal{P})"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi mathvariant="script">P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(\mathcal{P})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 的转移概率上的概率模型。给定状态、动作和奖励的历史 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="h_t=s_0a_0\dots r_{t-1}a_{t-1}s_t"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>h</mi><mi>t</mi></msub><mo>=</mo><msub><mi>s</mi><mn>0</mn></msub><msub><mi>a</mi><mn>0</mn></msub><mo>…</mo><msub><mi>r</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>a</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>s</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">h_t=s_0a_0\dots r_{t-1}a_{t-1}s_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">，我们可以计算一个 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathcal{P}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">P</mi></mrow><annotation encoding="application/x-tex">\mathcal{P}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span></span></span></span></span></span><span class="SemanticString"> 上的后验分布通过</span></span></p></div><p id="https://www.notion.so/39cbaac13411462d8033431e83b2ec86" class="Equation" data-latex="P(\mathcal{P}|h_t)\propto P(\mathcal{P})P(h_t|\mathcal{P})=P(\mathcal{P})\prod_{\tau=0}^t\mathcal{P}(s_{\tau+1}|s_\tau,a_\tau)."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi mathvariant="script">P</mi><mi mathvariant="normal">∣</mi><msub><mi>h</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mo>∝</mo><mi>P</mi><mo stretchy="false">(</mo><mi mathvariant="script">P</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mi>t</mi></msub><mi mathvariant="normal">∣</mi><mi mathvariant="script">P</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi mathvariant="script">P</mi><mo stretchy="false">)</mo><munderover><mo>∏</mo><mrow><mi>τ</mi><mo>=</mo><mn>0</mn></mrow><mi>t</mi></munderover><mi mathvariant="script">P</mi><mo stretchy="false">(</mo><msub><mi>s</mi><mrow><mi>τ</mi><mo>+</mo><mn>1</mn></mrow></msub><mi mathvariant="normal">∣</mi><msub><mi>s</mi><mi>τ</mi></msub><mo separator="true">,</mo><msub><mi>a</mi><mi>τ</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">P(\mathcal{P}|h_t)\propto P(\mathcal{P})P(h_t|\mathcal{P})=P(\mathcal{P})\prod_{\tau=0}^t\mathcal{P}(s_{\tau+1}|s_\tau,a_\tau).</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="mclose">)</span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.047674em;vertical-align:-1.267113em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7805610000000003em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.1132em;">τ</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∏</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.1132em;">τ</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.1132em;">τ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.1132em;">τ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord">.</span></span></span></span></span></p><div id="https://www.notion.so/6bc5412e5a584f7a8557036cc8f552ee" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">为了把这个带回 MDP 形式化，我们考虑一个增强状态空间 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="S^+"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>S</mi><mo>+</mo></msup></mrow><annotation encoding="application/x-tex">S^+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.771331em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.771331em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">，包含整个历史，封装了当前状态和我们对于转移模型的信念。在增强状态空间 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="S^+"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>S</mi><mo>+</mo></msup></mrow><annotation encoding="application/x-tex">S^+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.771331em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.771331em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 中的转移由对当前的信念在 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathcal{P}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">P</mi></mrow><annotation encoding="application/x-tex">\mathcal{P}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span></span></span></span></span></span><span class="SemanticString"> 上进行积分得到</span></span></p></div><p id="https://www.notion.so/68e1a96ca64245f1958e1b2ef3588d37" class="Equation" data-latex="\mathcal{P}^+(h_{t+1}|h_t,a_t)=\int P(\mathcal{P}|h_t)\mathcal{P}(s_{t+1}|s_t,a_t)d\mathcal{P}."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="script">P</mi><mo>+</mo></msup><mo stretchy="false">(</mo><msub><mi>h</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mi mathvariant="normal">∣</mi><msub><mi>h</mi><mi>t</mi></msub><mo separator="true">,</mo><msub><mi>a</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo>∫</mo><mi>P</mi><mo stretchy="false">(</mo><mi mathvariant="script">P</mi><mi mathvariant="normal">∣</mi><msub><mi>h</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mi mathvariant="script">P</mi><mo stretchy="false">(</mo><msub><mi>s</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mi mathvariant="normal">∣</mi><msub><mi>s</mi><mi>t</mi></msub><mo separator="true">,</mo><msub><mi>a</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mi>d</mi><mi mathvariant="script">P</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\mathcal{P}^+(h_{t+1}|h_t,a_t)=\int P(\mathcal{P}|h_t)\mathcal{P}(s_{t+1}|s_t,a_t)d\mathcal{P}.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.071331em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.821331em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.22225em;vertical-align:-0.86225em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∫</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="mord">.</span></span></span></span></span></p><div id="https://www.notion.so/9695d7a3af3045bca7ef02ea13888def" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">BAMDP 也有可能包含未知的奖励分布（如见</span><span class="SemanticString"><a class="SemanticString__Fragment SemanticString__Fragment--Link" href="https://arxiv.org/abs/1910.08348">此文</a></span><span class="SemanticString">），其中更外一层的奖励分布上模型在 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="h_t"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>h</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">h_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 的基础上以和转移概率相同的方式更新。具体来说，如果我们有一个奖励分布上的先验 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="P(R)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(R)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">，那么在增强状态 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="h_t"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>h</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">h_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 中进行行动 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a_t"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">a_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 的奖励函数是</span></span></p></div><p id="https://www.notion.so/e2ff6f2b6847400faaf69976526c4196" class="Equation" data-latex="R^+(r|h_t,a_t,h_{t+1})=\int P(R|h_{t+1})R(r|s_t,a_t,s_{t+1})dR."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>R</mi><mo>+</mo></msup><mo stretchy="false">(</mo><mi>r</mi><mi mathvariant="normal">∣</mi><msub><mi>h</mi><mi>t</mi></msub><mo separator="true">,</mo><msub><mi>a</mi><mi>t</mi></msub><mo separator="true">,</mo><msub><mi>h</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mo>∫</mo><mi>P</mi><mo stretchy="false">(</mo><mi>R</mi><mi mathvariant="normal">∣</mi><msub><mi>h</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mi>R</mi><mo stretchy="false">(</mo><mi>r</mi><mi mathvariant="normal">∣</mi><msub><mi>s</mi><mi>t</mi></msub><mo separator="true">,</mo><msub><mi>a</mi><mi>t</mi></msub><mo separator="true">,</mo><msub><mi>s</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mi>R</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">R^+(r|h_t,a_t,h_{t+1})=\int P(R|h_{t+1})R(r|s_t,a_t,s_{t+1})dR.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.071331em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.821331em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.22225em;vertical-align:-0.86225em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∫</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="mord">.</span></span></span></span></span></p><div id="https://www.notion.so/83afd248e8ed42e0a4237f11ee4c4b36" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">将这些合并起来得到一个新的 MDP，有历史的状态空间 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="S^+"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>S</mi><mo>+</mo></msup></mrow><annotation encoding="application/x-tex">S^+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.771331em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.771331em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">，没有改变的动作空间 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="A"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">A</span></span></span></span></span></span><span class="SemanticString">，增强的转移模型 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathcal{P}^+"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="script">P</mi><mo>+</mo></msup></mrow><annotation encoding="application/x-tex">\mathcal{P}^+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.771331em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.771331em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">，增强的奖励分布 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="R^+"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>R</mi><mo>+</mo></msup></mrow><annotation encoding="application/x-tex">R^+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.771331em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.771331em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">，折扣因子 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\gamma"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05556em;">γ</span></span></span></span></span></span><span class="SemanticString"> 和时间视野 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="T"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span></span></span></span></span></span><span class="SemanticString">。在这个新的 MDP 中的最优动作给出了探索与利用间最优的权衡。</span></span></p></div><h2 id="https://www.notion.so/ebc5e7d7a24b44058a753d0afed261e7" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--2"><a class="Anchor" href="#https://www.notion.so/ebc5e7d7a24b44058a753d0afed261e7"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">DAD 的贝叶斯 RL 形式化</span></span></h2><div id="https://www.notion.so/9352fe9381034dc7ae8306dec353a3d8" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">在 </span><span class="SemanticString"><a class="SemanticString__Fragment SemanticString__Fragment--Link" href="https://patrickg1014.github.io/dad">DAD</a></span><span class="SemanticString"> 中，我们选择一系列设计 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\xi_1,\dots,\xi_T"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ξ</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>ξ</mi><mi>T</mi></msub></mrow><annotation encoding="application/x-tex">\xi_1,\dots,\xi_T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 目的是最大化关于感兴趣的潜在参数 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\theta"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span></span></span><span class="SemanticString"> 的期望信息增益。为了把 DAD 放入贝叶斯 RL 的设定下，我们首先将在观察到结果前选择的设计 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\xi_t"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ξ</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\xi_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 和动作 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a_{t-1}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">a_{t-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 联系在一起。时间标签的差异是必要的，因为 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\xi_t"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ξ</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\xi_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 是在 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="y_t"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">y_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 被观测到之前被选择的。由于观测分布 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p(y|\xi,\theta)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>y</mi><mi mathvariant="normal">∣</mi><mi>ξ</mi><mo separator="true">,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(y|\xi,\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 依赖于未知的 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\theta"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span></span></span><span class="SemanticString">，我们不是处在一个 MDP 中，而是一个 BAMDP 中。如前面小节所述，将 DAD 的状态空间视为历史 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="h_t=\xi_1y_1\dots\xi_ty_t"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>h</mi><mi>t</mi></msub><mo>=</mo><msub><mi>ξ</mi><mn>1</mn></msub><msub><mi>y</mi><mn>1</mn></msub><mo>…</mo><msub><mi>ξ</mi><mi>t</mi></msub><msub><mi>y</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">h_t=\xi_1y_1\dots\xi_ty_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 空间似乎是明智的。DAD 转移模型上的不确定性体现在 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\theta"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span></span></span><span class="SemanticString"> 的不确定性上。具体来说，我们有下面的历史状态的转移分布</span></span></p></div><p id="https://www.notion.so/e891de0e8a1443e19d5e203ba484e033" class="Equation" data-latex="p(h_{t+1}|h_t,\xi_{t+1})=\int p(\theta|h_t)p(y_{t+1}|\xi_{t+1},\theta)d\theta"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mi mathvariant="normal">∣</mi><msub><mi>h</mi><mi>t</mi></msub><mo separator="true">,</mo><msub><mi>ξ</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mo>∫</mo><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mi mathvariant="normal">∣</mi><msub><mi>h</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>y</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mi mathvariant="normal">∣</mi><msub><mi>ξ</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><mi>θ</mi><mo stretchy="false">)</mo><mi>d</mi><mi>θ</mi></mrow><annotation encoding="application/x-tex">p(h_{t+1}|h_t,\xi_{t+1})=\int p(\theta|h_t)p(y_{t+1}|\xi_{t+1},\theta)d\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.22225em;vertical-align:-0.86225em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∫</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span></span></p><div id="https://www.notion.so/71c1714799464e4cb3ec8b544dfd039a" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">类似于 BAMDP 转移概率，但现在用实验设计的符号进行表示。不像标准强化学习设定，DAD 中没有外部奖励。相反，奖励是按照收集到的关于 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\theta"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span></span></span><span class="SemanticString"> 的信息来定义的。具体来说，我们可以取增强后的状态上的奖励分布 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="R^+(r|h_t,a_t,h_{t+1})"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>R</mi><mo>+</mo></msup><mo stretchy="false">(</mo><mi>r</mi><mi mathvariant="normal">∣</mi><msub><mi>h</mi><mi>t</mi></msub><mo separator="true">,</mo><msub><mi>a</mi><mi>t</mi></msub><mo separator="true">,</mo><msub><mi>h</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R^+(r|h_t,a_t,h_{t+1})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.021331em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.771331em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 为一个表示从 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="h_t"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>h</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">h_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 移动到 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="h_{t+1}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>h</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">h_{t+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 得到的关于 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\theta"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span></span></span><span class="SemanticString"> 的信息增益的 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="h_{t+1}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>h</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">h_{t+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 的确定性函数。这个函数有熵减给出</span></span></p></div><p id="https://www.notion.so/b165f70d542b4df0acbd57b37e80d7e9" class="Equation" data-latex="R^+(h_t,a_t,h_{t+1})=H[p(\theta|h_t)]-H[p(\theta|h_{t+1})]."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>R</mi><mo>+</mo></msup><mo stretchy="false">(</mo><msub><mi>h</mi><mi>t</mi></msub><mo separator="true">,</mo><msub><mi>a</mi><mi>t</mi></msub><mo separator="true">,</mo><msub><mi>h</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi>H</mi><mo stretchy="false">[</mo><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mi mathvariant="normal">∣</mi><msub><mi>h</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>−</mo><mi>H</mi><mo stretchy="false">[</mo><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mi mathvariant="normal">∣</mi><msub><mi>h</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">R^+(h_t,a_t,h_{t+1})=H[p(\theta|h_t)]-H[p(\theta|h_{t+1})].</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.071331em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.821331em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.08125em;">H</span><span class="mopen">[</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.08125em;">H</span><span class="mopen">[</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose">]</span><span class="mord">.</span></span></span></span></span></p><div id="https://www.notion.so/2e0768c4829f41819d5ea9a938e275b7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">为了使 BAMDP 规范更完整，我们取 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\gamma=1"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>γ</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\gamma=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05556em;">γ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span></span><span class="SemanticString"> 并使用 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="T"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span></span></span></span></span></span><span class="SemanticString"> 的时间视野。这给出策略的目标函数</span></span></p></div><p id="https://www.notion.so/2469a492ba564b1c9dcdf43b8bb6aa76" class="Equation" data-latex="\mathcal{J}(\pi)=\mathbb{E}\left[\sum_{t=1}^Tr_t\right]=\mathbb{E}_{p(\theta)p(h_T|\theta,\pi)}\left[\sum_{t=1}^TH[p(\theta|h_{t-1})]-H[p(\theta|h_t)]\right]."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">J</mi><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="double-struck">E</mi><mrow><mo fence="true">[</mo><munderover><mo>∑</mo><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow><mi>T</mi></munderover><msub><mi>r</mi><mi>t</mi></msub><mo fence="true">]</mo></mrow><mo>=</mo><msub><mi mathvariant="double-struck">E</mi><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mi>T</mi></msub><mi mathvariant="normal">∣</mi><mi>θ</mi><mo separator="true">,</mo><mi>π</mi><mo stretchy="false">)</mo></mrow></msub><mrow><mo fence="true">[</mo><munderover><mo>∑</mo><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow><mi>T</mi></munderover><mi>H</mi><mo stretchy="false">[</mo><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mi mathvariant="normal">∣</mi><msub><mi>h</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>−</mo><mi>H</mi><mo stretchy="false">[</mo><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mi mathvariant="normal">∣</mi><msub><mi>h</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo fence="true">]</mo></mrow><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\mathcal{J}(\pi)=\mathbb{E}\left[\sum_{t=1}^Tr_t\right]=\mathbb{E}_{p(\theta)p(h_T|\theta,\pi)}\left[\sum_{t=1}^TH[p(\theta|h_{t-1})]-H[p(\theta|h_t)]\right].</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.18472em;">J</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mord"><span class="mord mathbb">E</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">[</span></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">]</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mord"><span class="mord"><span class="mord mathbb">E</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.34480000000000005em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="mclose mtight">)</span><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathdefault mtight">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em;"><span></span></span></span></span></span></span><span class="mord mtight">∣</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">π</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3551999999999999em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">[</span></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.08125em;">H</span><span class="mopen">[</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault" style="margin-right:0.08125em;">H</span><span class="mopen">[</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose">]</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">]</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">.</span></span></span></span></span></p><div id="https://www.notion.so/3a6e0476f16040a58af0278de27796d9" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">为了把这个目标函数与 DAD 中使用的目标联系起来，我们使用 DAD 论文中的定理 1，得出</span></span></p></div><p id="https://www.notion.so/ca64503487b1495b98c6305c90b76035" class="Equation" data-latex="\mathcal{J}(\pi)=\mathbb{E}_{p(\theta)p(h_T|\theta,\pi)}\left[\sum_{t=1}^TH[p(\theta|h_{t-1})]-H[p(\theta|h_t)]\right]\xlongequal{\mathrm{Theorem\;1}}\mathcal{I}_T(\pi)"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">J</mi><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi mathvariant="double-struck">E</mi><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mi>T</mi></msub><mi mathvariant="normal">∣</mi><mi>θ</mi><mo separator="true">,</mo><mi>π</mi><mo stretchy="false">)</mo></mrow></msub><mrow><mo fence="true">[</mo><munderover><mo>∑</mo><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow><mi>T</mi></munderover><mi>H</mi><mo stretchy="false">[</mo><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mi mathvariant="normal">∣</mi><msub><mi>h</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>−</mo><mi>H</mi><mo stretchy="false">[</mo><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mi mathvariant="normal">∣</mi><msub><mi>h</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo fence="true">]</mo></mrow><mover><mo stretchy="true">=</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi mathvariant="normal">T</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">m</mi><mtext>  </mtext><mn>1</mn></mrow></mpadded></mover><msub><mi mathvariant="script">I</mi><mi>T</mi></msub><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{J}(\pi)=\mathbb{E}_{p(\theta)p(h_T|\theta,\pi)}\left[\sum_{t=1}^TH[p(\theta|h_{t-1})]-H[p(\theta|h_t)]\right]\xlongequal{\mathrm{Theorem\;1}}\mathcal{I}_T(\pi)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.18472em;">J</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mord"><span class="mord"><span class="mord mathbb">E</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.34480000000000005em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="mclose mtight">)</span><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathdefault mtight">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em;"><span></span></span></span></span></span></span><span class="mord mtight">∣</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">π</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3551999999999999em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">[</span></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.08125em;">H</span><span class="mopen">[</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault" style="margin-right:0.08125em;">H</span><span class="mopen">[</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose">]</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">]</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel x-arrow"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.014108em;"><span style="top:-3.228em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">T</span><span class="mord mathrm mtight">h</span><span class="mord mathrm mtight">e</span><span class="mord mathrm mtight">o</span><span class="mord mathrm mtight">r</span><span class="mord mathrm mtight">e</span><span class="mord mathrm mtight">m</span><span class="mspace mtight" style="margin-right:0.3252777777777778em;"></span><span class="mord mathrm mtight">1</span></span></span></span></span><span class="svg-align" style="top:-2.7830000000000004em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.334em;min-width:0.888em;"><svg width='400em' height='0.334em' viewBox='0 0 400000 334' preserveAspectRatio='xMinYMin slice'><path d='M0 50 h400000 v40H0z m0 194h40000v40H0z
 M0 50 h400000 v40H0z m0 194h40000v40H0z'/></svg></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mclose">)</span></span></span></span></span></p><div id="https://www.notion.so/8d00dea0bbca40e18a14b43d2f171b0b" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">其中</span></span></p></div><p id="https://www.notion.so/ebe86488578443ad842b8b9424212291" class="Equation" data-latex="\mathcal{I}_T(\pi)=\mathbb{E}_{p(\theta)p(h_T|\theta,\pi)}\left[\log\frac{p(h_T|\theta,\pi)}{\mathbb{E}_{p(\theta&#x27;)}p(h_T|\theta&#x27;,\pi)}\right]."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">I</mi><mi>T</mi></msub><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi mathvariant="double-struck">E</mi><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mi>T</mi></msub><mi mathvariant="normal">∣</mi><mi>θ</mi><mo separator="true">,</mo><mi>π</mi><mo stretchy="false">)</mo></mrow></msub><mrow><mo fence="true">[</mo><mi>log</mi><mo>⁡</mo><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mi>T</mi></msub><mi mathvariant="normal">∣</mi><mi>θ</mi><mo separator="true">,</mo><mi>π</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi mathvariant="double-struck">E</mi><mrow><mi>p</mi><mo stretchy="false">(</mo><msup><mi>θ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">)</mo></mrow></msub><mi>p</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mi>T</mi></msub><mi mathvariant="normal">∣</mi><msup><mi>θ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo separator="true">,</mo><mi>π</mi><mo stretchy="false">)</mo></mrow></mfrac><mo fence="true">]</mo></mrow><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\mathcal{I}_T(\pi)=\mathbb{E}_{p(\theta)p(h_T|\theta,\pi)}\left[\log\frac{p(h_T|\theta,\pi)}{\mathbb{E}_{p(\theta&#x27;)}p(h_T|\theta&#x27;,\pi)}\right].</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.4912em;vertical-align:-1.0412em;"></span><span class="mord"><span class="mord"><span class="mord mathbb">E</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.34480000000000005em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="mclose mtight">)</span><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathdefault mtight">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em;"><span></span></span></span></span></span></span><span class="mord mtight">∣</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">π</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3551999999999999em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathbb">E</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.34480000000000005em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3551999999999999em;"><span></span></span></span></span></span></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6778919999999999em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0412em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">.</span></span></span></span></span></p><div id="https://www.notion.so/5451ce262cae47009633a17cc38f955d" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">总之，我们可以把 DAD 解决的问题投射为一个 BAMDP。我们将设计识别为动作，实验历史识别为增强后的状态，我们使用概率模型给出一个在这些状态上的自然的转移分布，我们引入了非随机的一步信息增益的奖励，我们设定 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\gamma=1"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>γ</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\gamma=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05556em;">γ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span></span><span class="SemanticString"> 并通常假定有限次实验迭代 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="T"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span></span></span></span></span></span><span class="SemanticString">。</span></span></p></div><h2 id="https://www.notion.so/f58e4d62fc044907abb927b9378a1ba9" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--2"><a class="Anchor" href="#https://www.notion.so/f58e4d62fc044907abb927b9378a1ba9"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">什么使实验设计问题不同？</span></span></h2><div id="https://www.notion.so/a05de765c02d4227879a865471621b12" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">在建立了序列贝叶斯实验设计和贝叶斯 RL 间的理论联系之后，有人可能自然要问当通用的贝叶斯强化学习算法适用时，是否有理由开发专门的实验设计算法。首先，我们关注贝叶斯实验设计问题的奖励结构。奖励 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="r_t=H[p(\theta|h_{t-1})]-H[p(\theta|h_t)]"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mi>t</mi></msub><mo>=</mo><mi>H</mi><mo stretchy="false">[</mo><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mi mathvariant="normal">∣</mi><msub><mi>h</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>−</mo><mi>H</mi><mo stretchy="false">[</mo><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mi mathvariant="normal">∣</mi><msub><mi>h</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">r_t=H[p(\theta|h_{t-1})]-H[p(\theta|h_t)]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.08125em;">H</span><span class="mopen">[</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.08125em;">H</span><span class="mopen">[</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose">]</span></span></span></span></span></span><span class="SemanticString"> 通常是难解的，需要 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\theta"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span></span></span><span class="SemanticString"> 上的贝叶斯推断。不是尝试去估计这个奖励，DAD 提出了策略 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\pi"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span></span></span></span></span></span><span class="SemanticString"> 下的总期望信息增益的 sPCE 下界，即</span></span></p></div><p id="https://www.notion.so/f05ac63cab8c4ce4aded8753e7b99c1c" class="Equation" data-latex="\mathcal{I}_T(\pi)\ge\mathcal{L}_T(\pi,L)=\mathbb{E}_{p(\theta_0)p(h_T|\theta_0,\pi)p(\theta_{1:L})}\left[\log\cfrac{p(h_T|\theta_0,\pi)}{\frac{1}{L+1}\sum_{\ell=0}^Lp(h_T|\theta_\ell,\pi)}\right]."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">I</mi><mi>T</mi></msub><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">)</mo><mo>≥</mo><msub><mi mathvariant="script">L</mi><mi>T</mi></msub><mo stretchy="false">(</mo><mi>π</mi><mo separator="true">,</mo><mi>L</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi mathvariant="double-struck">E</mi><mrow><mi>p</mi><mo stretchy="false">(</mo><msub><mi>θ</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mi>T</mi></msub><mi mathvariant="normal">∣</mi><msub><mi>θ</mi><mn>0</mn></msub><mo separator="true">,</mo><mi>π</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>θ</mi><mrow><mn>1</mn><mo>:</mo><mi>L</mi></mrow></msub><mo stretchy="false">)</mo></mrow></msub><mrow><mo fence="true">[</mo><mi>log</mi><mo>⁡</mo><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mi>T</mi></msub><mi mathvariant="normal">∣</mi><msub><mi>θ</mi><mn>0</mn></msub><mo separator="true">,</mo><mi>π</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac><mn>1</mn><mrow><mi>L</mi><mo>+</mo><mn>1</mn></mrow></mfrac><munderover><mo>∑</mo><mrow><mi mathvariant="normal">ℓ</mi><mo>=</mo><mn>0</mn></mrow><mi>L</mi></munderover><mi>p</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mi>T</mi></msub><mi mathvariant="normal">∣</mi><msub><mi>θ</mi><mi mathvariant="normal">ℓ</mi></msub><mo separator="true">,</mo><mi>π</mi><mo stretchy="false">)</mo></mrow></mfrac><mo fence="true">]</mo></mrow><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\mathcal{I}_T(\pi)\ge\mathcal{L}_T(\pi,L)=\mathbb{E}_{p(\theta_0)p(h_T|\theta_0,\pi)p(\theta_{1:L})}\left[\log\cfrac{p(h_T|\theta_0,\pi)}{\frac{1}{L+1}\sum_{\ell=0}^Lp(h_T|\theta_\ell,\pi)}\right].</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathcal">L</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">L</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.024562em;vertical-align:-1.274562em;"></span><span class="mord"><span class="mord"><span class="mord mathbb">E</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.34480000000000005em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mclose mtight">)</span><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathdefault mtight">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em;"><span></span></span></span></span></span></span><span class="mord mtight">∣</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">π</span><span class="mclose mtight">)</span><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567071428571427em;margin-left:-0.02778em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mrel mtight">:</span><span class="mord mathdefault mtight">L</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em;"><span></span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3551999999999999em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">[</span></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5899999999999999em;"><span style="top:-2.128769em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.403331em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">ℓ</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.74em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.274562em;"><span></span></span></span></span></span><span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">]</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">.</span></span></span></span></span></p><div id="https://www.notion.so/5f66de1b8d3e442c8db05da35ce57e4f" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">有趣的是，有一个在 RL 框架下解释 sPCE 目标的方式。首先，我们使用</span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">根采样</em></span><span class="SemanticString">（root sampling）来一起采样 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\theta_0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\theta_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 和 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="h_T"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>h</mi><mi>T</mi></msub></mrow><annotation encoding="application/x-tex">h_T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">。我们也固定对照量 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\theta_{1:L}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mrow><mn>1</mn><mo>:</mo><mi>L</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\theta_{1:L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mrel mtight">:</span><span class="mord mathdefault mtight">L</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">。最后，我们使用替代奖励</span></span></p></div><p id="https://www.notion.so/575e2c27f16e4f8e9b053cfc08c54393" class="Equation" data-latex="\tilde{r}_t=\log\cfrac{p(h_t|\theta_0,\pi)}{\frac{1}{L+1}\sum_{\ell=0}^Lp(h_t|\theta_\ell,\pi)}-\log\cfrac{p(h_{t-1}|\theta_0,\pi)}{\frac{1}{L+1}\sum_{\ell=0}^Lp(h_{t-1}|\theta_\ell,\pi)}."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>r</mi><mo>~</mo></mover><mi>t</mi></msub><mo>=</mo><mi>log</mi><mo>⁡</mo><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mi>t</mi></msub><mi mathvariant="normal">∣</mi><msub><mi>θ</mi><mn>0</mn></msub><mo separator="true">,</mo><mi>π</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac><mn>1</mn><mrow><mi>L</mi><mo>+</mo><mn>1</mn></mrow></mfrac><munderover><mo>∑</mo><mrow><mi mathvariant="normal">ℓ</mi><mo>=</mo><mn>0</mn></mrow><mi>L</mi></munderover><mi>p</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mi>t</mi></msub><mi mathvariant="normal">∣</mi><msub><mi>θ</mi><mi mathvariant="normal">ℓ</mi></msub><mo separator="true">,</mo><mi>π</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mi>log</mi><mo>⁡</mo><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mi mathvariant="normal">∣</mi><msub><mi>θ</mi><mn>0</mn></msub><mo separator="true">,</mo><mi>π</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac><mn>1</mn><mrow><mi>L</mi><mo>+</mo><mn>1</mn></mrow></mfrac><munderover><mo>∑</mo><mrow><mi mathvariant="normal">ℓ</mi><mo>=</mo><mn>0</mn></mrow><mi>L</mi></munderover><mi>p</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mi mathvariant="normal">∣</mi><msub><mi>θ</mi><mi mathvariant="normal">ℓ</mi></msub><mo separator="true">,</mo><mi>π</mi><mo stretchy="false">)</mo></mrow></mfrac><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\tilde{r}_t=\log\cfrac{p(h_t|\theta_0,\pi)}{\frac{1}{L+1}\sum_{\ell=0}^Lp(h_t|\theta_\ell,\pi)}-\log\cfrac{p(h_{t-1}|\theta_0,\pi)}{\frac{1}{L+1}\sum_{\ell=0}^Lp(h_{t-1}|\theta_\ell,\pi)}.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8178599999999999em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">r</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.864562em;vertical-align:-1.274562em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5899999999999999em;"><span style="top:-2.128769em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.403331em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">ℓ</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.74em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.274562em;"><span></span></span></span></span></span><span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.864562em;vertical-align:-1.274562em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5899999999999999em;"><span style="top:-2.128769em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.403331em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">ℓ</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.74em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.274562em;"><span></span></span></span></span></span><span></span></span><span class="mord">.</span></span></span></span></span></p><div id="https://www.notion.so/80fc201319334314804c9700aeb8da4a" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">由于这些奖励依赖于 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\theta_0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\theta_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">，如果我们只以 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="h_t"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>h</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">h_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 为条件的话我们可以把它们当作随机化的奖励。</span></span></p></div><div id="https://www.notion.so/d15af316125642ed8db95e0fd638a692" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">这些奖励的一个重要特征是，尽管不可解，替代的 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathcal{L}_T(\pi,L)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">L</mi><mi>T</mi></msub><mo stretchy="false">(</mo><mi>π</mi><mo separator="true">,</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{L}_T(\pi,L)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathcal">L</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">L</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 是关于设计 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="(\xi_t)_{t=1}^T"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ξ</mi><mi>t</mi></msub><msubsup><mo stretchy="false">)</mo><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow><mi>T</mi></msubsup></mrow><annotation encoding="application/x-tex">(\xi_t)_{t=1}^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.4518920000000004em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24810799999999997em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 和观测 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="(y_t)_{t=1}^T"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>y</mi><mi>t</mi></msub><msubsup><mo stretchy="false">)</mo><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow><mi>T</mi></msubsup></mrow><annotation encoding="application/x-tex">(y_t)_{t=1}^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.4518920000000004em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24810799999999997em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 可微分的。在 DAD 的最简单形式中，我们进一步假设 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="y_t"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">y_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 和 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\xi_t"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ξ</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\xi_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 之间的以一种可重参数化的方式封装的可微分关系来采样 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p(y|\theta,\xi)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>y</mi><mi mathvariant="normal">∣</mi><mi>θ</mi><mo separator="true">,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(y|\theta,\xi)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">。具体来说，例如我们可能有 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="y|\theta,\xi=\mu(\theta,\xi)+\sigma(\theta,\xi)\epsilon"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mi mathvariant="normal">∣</mi><mi>θ</mi><mo separator="true">,</mo><mi>ξ</mi><mo>=</mo><mi>μ</mi><mo stretchy="false">(</mo><mi>θ</mi><mo separator="true">,</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo>+</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>θ</mi><mo separator="true">,</mo><mi>ξ</mi><mo stretchy="false">)</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">y|\theta,\xi=\mu(\theta,\xi)+\sigma(\theta,\xi)\epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">μ</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">σ</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span><span class="mord mathdefault">ϵ</span></span></span></span></span></span><span class="SemanticString"> 其中 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\epsilon\sim N(0,1)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϵ</mi><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\epsilon\sim N(0,1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">ϵ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 且 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mu"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">μ</span></span></span></span></span></span><span class="SemanticString"> 和 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\sigma"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">σ</span></span></span></span></span></span><span class="SemanticString"> 都是可微分函数。这些假设的结果是我们可以直接关于根据公式 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\xi_t=\pi_\phi(h_{t-1})"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ξ</mi><mi>t</mi></msub><mo>=</mo><msub><mi>π</mi><mi>ϕ</mi></msub><mo stretchy="false">(</mo><msub><mi>h</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\xi_t=\pi_\phi(h_{t-1})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">ϕ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 生成设计 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="(\xi_t)_{t=1}^T"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ξ</mi><mi>t</mi></msub><msubsup><mo stretchy="false">)</mo><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow><mi>T</mi></msubsup></mrow><annotation encoding="application/x-tex">(\xi_t)_{t=1}^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.4518920000000004em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24810799999999997em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 的策略网络 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\pi_\phi"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>π</mi><mi>ϕ</mi></msub></mrow><annotation encoding="application/x-tex">\pi_\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">ϕ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 的参数 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\phi"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">ϕ</span></span></span></span></span></span><span class="SemanticString"> 求替代目标 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathcal{L}_T(\pi,L)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">L</mi><mi>T</mi></msub><mo stretchy="false">(</mo><mi>π</mi><mo separator="true">,</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{L}_T(\pi,L)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathcal">L</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">L</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 微分。DAD 直接通过在 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathcal{L}_T(\pi,L)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">L</mi><mi>T</mi></msub><mo stretchy="false">(</mo><mi>π</mi><mo separator="true">,</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{L}_T(\pi,L)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathcal">L</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">L</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 上的梯度下降优化策略 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\pi_\phi"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>π</mi><mi>ϕ</mi></msub></mrow><annotation encoding="application/x-tex">\pi_\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">ϕ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">。</span></span></p></div><div id="https://www.notion.so/77c584e0d187439694a1b10833539f06" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">因此，DAD 可以用 RL 的语言描述为一个直接策略优化方法。尽管</span><span class="SemanticString"><a class="SemanticString__Fragment SemanticString__Fragment--Link" href="https://arxiv.org/abs/1906.06062">直接策略优化方法</a></span><span class="SemanticString">在 RL 中被使用，它们与标准相去甚远，</span><span class="SemanticString"><a class="SemanticString__Fragment SemanticString__Fragment--Link" href="https://link.springer.com/article/10.1007/BF00992698">Q-learning</a></span><span class="SemanticString"> 和 </span><span class="SemanticString"><a class="SemanticString__Fragment SemanticString__Fragment--Link" href="https://proceedings.neurips.cc/paper/1999/file/6449f44a102fde848669bdd9eb6b76fa-Paper.pdf">actor-critic</a></span><span class="SemanticString"> 等方法更占主导地位。这也许是因为 RL 通常不假设奖励函数是可微分的：例如，一个真实环境中的奖励很少有梯度信息。也可能是因为离散动作问题更是关注的焦点。</span></span></p></div><div id="https://www.notion.so/4d1356b5a47d4e13bf5b267698531084" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">DAD 也不同于</span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">贝叶斯</em></span><span class="SemanticString"> RL 的许多方法在于它不去估计后验 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p(\theta|h_t,\pi)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mi mathvariant="normal">∣</mi><msub><mi>h</mi><mi>t</mi></msub><mo separator="true">,</mo><mi>π</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(\theta|h_t,\pi)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">。在贝叶斯 RL 中，这些后验分布被称为</span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">信念状态</em></span><span class="SemanticString">。许多解决贝叶斯 RL 问题的方法使用信念状态的估计。相反，DAD依赖于一种更接近与根采样的方法。这也是 DAD 和之前的</span><span class="SemanticString"><a class="SemanticString__Fragment SemanticString__Fragment--Link" href="https://arxiv.org/abs/1604.08320">非贪婪的序列贝叶斯实验设计</a></span><span class="SemanticString">间的不同。</span></span></p></div><h2 id="https://www.notion.so/f520cbd885594e28b790d8f170163f7d" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--2"><a class="Anchor" href="#https://www.notion.so/f520cbd885594e28b790d8f170163f7d"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">DAD 的新的目标函数</span></span></h2><div id="https://www.notion.so/a8344f28700c4fc1b55e4cdb85bd451b" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">在贝叶斯 RL 的框架中理解 DAD 自然会引发一个问题，即一般的 DAD 方法是否可以应用于非信息增益的目标函数（奖励）。前面的讨论表明，使用根采样所以对于 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\theta"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span></span></span><span class="SemanticString"> 的依赖是可能的，我们可以考虑这个形式的奖励</span></span></p></div><p id="https://www.notion.so/42dfcb0b42dc4e839f48937a71133f99" class="Equation" data-latex="r_t^{\mathrm{general}}=R(\theta,h_t,\epsilon_t)"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>r</mi><mi>t</mi><mrow><mi mathvariant="normal">g</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">l</mi></mrow></msubsup><mo>=</mo><mi>R</mi><mo stretchy="false">(</mo><mi>θ</mi><mo separator="true">,</mo><msub><mi>h</mi><mi>t</mi></msub><mo separator="true">,</mo><msub><mi>ϵ</mi><mi>t</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r_t^{\mathrm{general}}=R(\theta,h_t,\epsilon_t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2127719999999997em;vertical-align:-0.24575599999999997em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9670159999999999em;"><span style="top:-2.454244em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight" style="margin-right:0.01389em;">g</span><span class="mord mathrm mtight">e</span><span class="mord mathrm mtight">n</span><span class="mord mathrm mtight">e</span><span class="mord mathrm mtight">r</span><span class="mord mathrm mtight">a</span><span class="mord mathrm mtight">l</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24575599999999997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">ϵ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p><div id="https://www.notion.so/477b9ba16cef4b1dad4ab4ac135a7631" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">其中 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="R"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span></span></span></span></span></span><span class="SemanticString"> 是一个一致的可微分函数，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\epsilon_t"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ϵ</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\epsilon_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">ϵ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 是一个独立的噪声随机变量。显然，信息增益奖励 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="r_t"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">r_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 满足这个模式，只是 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="h_t"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>h</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">h_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 的函数。将这个可微分奖励函数与重参数化假设合并起来意味着一般的奖励</span></span></p></div><p id="https://www.notion.so/039d6f6798a0461bbc19e832d9cd00b5" class="Equation" data-latex="\mathcal{J}^{\mathrm{general}}(\pi)=\mathbb{E}_{p(\theta)p(h_T)p(\epsilon_{1:T})}\left[\sum_{t=1}^Tr_t^{\mathrm{general}}\right]"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="script">J</mi><mrow><mi mathvariant="normal">g</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">l</mi></mrow></msup><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi mathvariant="double-struck">E</mi><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mi>T</mi></msub><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>ϵ</mi><mrow><mn>1</mn><mo>:</mo><mi>T</mi></mrow></msub><mo stretchy="false">)</mo></mrow></msub><mrow><mo fence="true">[</mo><munderover><mo>∑</mo><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow><mi>T</mi></munderover><msubsup><mi>r</mi><mi>t</mi><mrow><mi mathvariant="normal">g</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">l</mi></mrow></msubsup><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\mathcal{J}^{\mathrm{general}}(\pi)=\mathbb{E}_{p(\theta)p(h_T)p(\epsilon_{1:T})}\left[\sum_{t=1}^Tr_t^{\mathrm{general}}\right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.149108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathcal" style="margin-right:0.18472em;">J</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em;"><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight" style="margin-right:0.01389em;">g</span><span class="mord mathrm mtight">e</span><span class="mord mathrm mtight">n</span><span class="mord mathrm mtight">e</span><span class="mord mathrm mtight">r</span><span class="mord mathrm mtight">a</span><span class="mord mathrm mtight">l</span></span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mord"><span class="mord"><span class="mord mathbb">E</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.34480000000000005em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="mclose mtight">)</span><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathdefault mtight">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em;"><span></span></span></span></span></span></span><span class="mclose mtight">)</span><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathdefault mtight">ϵ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mrel mtight">:</span><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em;"><span></span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3551999999999999em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">[</span></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9670159999999999em;"><span style="top:-2.454244em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight" style="margin-right:0.01389em;">g</span><span class="mord mathrm mtight">e</span><span class="mord mathrm mtight">n</span><span class="mord mathrm mtight">e</span><span class="mord mathrm mtight">r</span><span class="mord mathrm mtight">a</span><span class="mord mathrm mtight">l</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24575599999999997em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">]</span></span></span></span></span></span></span></p><div id="https://www.notion.so/19512b99de99486ca83b53ba2b3fbcff" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">可以通过直接策略梯度关于 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\pi"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span></span></span></span></span></span><span class="SemanticString"> 直接被优化。在实验设计情境中，这就为 DAD 的两个相对简单的扩展打开了大门。例如，我们可以给每个设计分配一个（可微分的）开销。假设我们用负的设计开销的和增强原始的期望信息增益目标。使用 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\lambda"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">λ</span></span></span></span></span></span><span class="SemanticString"> 来权衡开销和信息，我们得到</span></span></p></div><p id="https://www.notion.so/0b4dd63c355c484b902a43fbc71762ff" class="Equation" data-latex="\mathcal{J}^{\mathrm{costed}}(\pi)=\mathcal{I}_T(\pi)-\lambda\mathbb{E}\left[\sum_{t=1}^TC(\xi_t)\right]"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="script">J</mi><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi></mrow></msup><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi mathvariant="script">I</mi><mi>T</mi></msub><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">)</mo><mo>−</mo><mi>λ</mi><mi mathvariant="double-struck">E</mi><mrow><mo fence="true">[</mo><munderover><mo>∑</mo><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow><mi>T</mi></munderover><mi>C</mi><mo stretchy="false">(</mo><msub><mi>ξ</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\mathcal{J}^{\mathrm{costed}}(\pi)=\mathcal{I}_T(\pi)-\lambda\mathbb{E}\left[\sum_{t=1}^TC(\xi_t)\right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.149108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathcal" style="margin-right:0.18472em;">J</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">c</span><span class="mord mathrm mtight">o</span><span class="mord mathrm mtight">s</span><span class="mord mathrm mtight">t</span><span class="mord mathrm mtight">e</span><span class="mord mathrm mtight">d</span></span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mord mathdefault">λ</span><span class="mord"><span class="mord mathbb">E</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">[</span></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">]</span></span></span></span></span></span></span></p><div id="https://www.notion.so/41c96538fcc84d1a8f1c8724b0eb3390" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">我们可以使用本质上和 DAD 相同的方法解决该问题。第二，我们可以考虑最终后验分布的质量的不同度量。例如，一维的 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\theta"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span></span></span><span class="SemanticString">，我们可能比起降低后验熵对于降低后验方差更感兴趣。我们可以取奖励函数为</span></span></p></div><p id="https://www.notion.so/cb1b977ed5e14c099a7d8c296b404bd4" class="Equation" data-latex="r_t^{\mathrm{variance}}=\mathrm{Var}_{p(\theta|h_{t-1})}[\theta]-\mathrm{Var}_{p(\theta|h_t)}[\theta]."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>r</mi><mi>t</mi><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">e</mi></mrow></msubsup><mo>=</mo><msub><mrow><mi mathvariant="normal">V</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi></mrow><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mi mathvariant="normal">∣</mi><msub><mi>h</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">[</mo><mi>θ</mi><mo stretchy="false">]</mo><mo>−</mo><msub><mrow><mi mathvariant="normal">V</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi></mrow><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mi mathvariant="normal">∣</mi><msub><mi>h</mi><mi>t</mi></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">[</mo><mi>θ</mi><mo stretchy="false">]</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">r_t^{\mathrm{variance}}=\mathrm{Var}_{p(\theta|h_{t-1})}[\theta]-\mathrm{Var}_{p(\theta|h_t)}[\theta].</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1275019999999998em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8805019999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight" style="margin-right:0.01389em;">v</span><span class="mord mathrm mtight">a</span><span class="mord mathrm mtight">r</span><span class="mord mathrm mtight">i</span><span class="mord mathrm mtight">a</span><span class="mord mathrm mtight">n</span><span class="mord mathrm mtight">c</span><span class="mord mathrm mtight">e</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1052em;vertical-align:-0.3551999999999999em;"></span><span class="mord"><span class="mord"><span class="mord mathrm" style="margin-right:0.01389em;">V</span><span class="mord mathrm">a</span><span class="mord mathrm">r</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.34480000000000005em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="mord mtight">∣</span><span class="mord mtight"><span class="mord mathdefault mtight">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173142857142857em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.20252142857142857em;"><span></span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3551999999999999em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.1052em;vertical-align:-0.3551999999999999em;"></span><span class="mord"><span class="mord"><span class="mord mathrm" style="margin-right:0.01389em;">V</span><span class="mord mathrm">a</span><span class="mord mathrm">r</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.34480000000000005em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="mord mtight">∣</span><span class="mord mtight"><span class="mord mathdefault mtight">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.29634285714285713em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3551999999999999em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mclose">]</span><span class="mord">.</span></span></span></span></span></p><div id="https://www.notion.so/63b9c4394f514eafa54d0738733e0c6e" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">尽管有确定的原因可以解释为什么上方法在理论上更合理，但使用后验分布的不同函数作为奖励信号确实相对自然地符合 DAD 框架。剩下的问题是，是否可以像 DAD 使用 sPCE 来估计信息增益那样有效地估计该函数。对于方差，我们有</span></span></p></div><p id="https://www.notion.so/e0b244bca68c4515a628ea02e9e7ab6e" class="Equation" data-latex="\mathbb{E}_{p(\theta)p(h_T|\theta,\pi)}\left[\sum_{t=1}^Tr_t^{\mathrm{variance}}\right]\ge\mathrm{Var}_{p(\theta)}[\theta]-\mathbb{E}_{p(\theta)p(h_T|\theta,\pi)}[(\theta-f_{\phi&#x27;}(h_T))^2]"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="double-struck">E</mi><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mi>T</mi></msub><mi mathvariant="normal">∣</mi><mi>θ</mi><mo separator="true">,</mo><mi>π</mi><mo stretchy="false">)</mo></mrow></msub><mrow><mo fence="true">[</mo><munderover><mo>∑</mo><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow><mi>T</mi></munderover><msubsup><mi>r</mi><mi>t</mi><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">e</mi></mrow></msubsup><mo fence="true">]</mo></mrow><mo>≥</mo><msub><mrow><mi mathvariant="normal">V</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi></mrow><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">[</mo><mi>θ</mi><mo stretchy="false">]</mo><mo>−</mo><msub><mi mathvariant="double-struck">E</mi><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>h</mi><mi>T</mi></msub><mi mathvariant="normal">∣</mi><mi>θ</mi><mo separator="true">,</mo><mi>π</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>θ</mi><mo>−</mo><msub><mi>f</mi><msup><mi>ϕ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></msub><mo stretchy="false">(</mo><msub><mi>h</mi><mi>T</mi></msub><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{E}_{p(\theta)p(h_T|\theta,\pi)}\left[\sum_{t=1}^Tr_t^{\mathrm{variance}}\right]\ge\mathrm{Var}_{p(\theta)}[\theta]-\mathbb{E}_{p(\theta)p(h_T|\theta,\pi)}[(\theta-f_{\phi&#x27;}(h_T))^2]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mord"><span class="mord"><span class="mord mathbb">E</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.34480000000000005em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="mclose mtight">)</span><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathdefault mtight">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em;"><span></span></span></span></span></span></span><span class="mord mtight">∣</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">π</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3551999999999999em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">[</span></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8805019999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight" style="margin-right:0.01389em;">v</span><span class="mord mathrm mtight">a</span><span class="mord mathrm mtight">r</span><span class="mord mathrm mtight">i</span><span class="mord mathrm mtight">a</span><span class="mord mathrm mtight">n</span><span class="mord mathrm mtight">c</span><span class="mord mathrm mtight">e</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">]</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1052em;vertical-align:-0.3551999999999999em;"></span><span class="mord"><span class="mord"><span class="mord mathrm" style="margin-right:0.01389em;">V</span><span class="mord mathrm">a</span><span class="mord mathrm">r</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.34480000000000005em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3551999999999999em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.1052em;vertical-align:-0.3551999999999999em;"></span><span class="mord"><span class="mord"><span class="mord mathbb">E</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.34480000000000005em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="mclose mtight">)</span><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathdefault mtight">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em;"><span></span></span></span></span></span></span><span class="mord mtight">∣</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">π</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3551999999999999em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.150216em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">ϕ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></span></p><div id="https://www.notion.so/10f9527858554adc80abb38b735a094d" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">其中 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="f_{\phi&#x27;}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><msup><mi>ϕ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></msub></mrow><annotation encoding="application/x-tex">f_{\phi&#x27;}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">ϕ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 是一个可学习的函数。注意其与 </span><span class="SemanticString"><a class="SemanticString__Fragment SemanticString__Fragment--Link" href="http://aivalley.com/Papers/MI_NIPS_final.pdf">Barber-Agakov 界</a></span><span class="SemanticString">的相似性。</span></span></p></div><h2 id="https://www.notion.so/b52aa668d52c4154b6aa5e657c642ce1" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--2"><a class="Anchor" href="#https://www.notion.so/b52aa668d52c4154b6aa5e657c642ce1"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">用于贝叶斯实验设计的 RL 算法</span></span></h2><div id="https://www.notion.so/1ded105fc89e452ab3d3e1ffb1c23633" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">总之，在序列贝叶斯实验设计之间建立正式的联系，为利用大量关于贝叶斯 RL 和控制理论的文献来提高我们计划序列实验的能力提供了可能性。虽然 DAD 的直接策略优化方法非常有效，但当这种训练方法开始失效时，理解与 RL 的联系应该会帮助我们。将现有的贝叶斯 RL 算法应用于实验设计是一个令人兴奋的新研究领域。（</span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">2022 年更新</em></span><span class="SemanticString">：事实上，这是最近几篇论文探索的角度。）</span></span></p></div><div id="https://www.notion.so/69d8dde2f67149858fb2d3f44b60c91e" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">对于 DAD 来说，这些见解可能有用的一个可能的困难是在长期实验中。为了有效地规划长期实验，DAD 模拟了数千种可能的实验轨迹。然而，这种模拟的效率有可能会随着 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="T"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span></span></span></span></span></span><span class="SemanticString"> 增加而下降。DAD 非常需要数据：它在每个梯度步骤中重新模拟全新的轨迹。这避免了训练数据过时的问题，但也增加了训练成本。</span></span></p></div><div id="https://www.notion.so/d2155ce20f2946679111c0e585b5e746" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">同样可以想象的是，在某些情况下，不可能为未来的所有可能情况作出计划。RL 的类比是一个非常随机的环境，在游戏开始时从一个长列表中随机选择一个游戏。因此，智能体必须首先发现自己在玩哪种游戏，然后才能成功地玩下去。如果所有计划都是预先进行的，那么 RL 智能体必须在开始真实环境之前学习如何玩好每一款游戏。另一种方法是引入一些真实数据，并在进行过程中对策略进行重新训练。在 RL 情境中，这意味着在知道如何玩游戏之前就能发现玩家正在玩哪款游戏，这可以通过更简单的策略实现。一旦有了这个发现，我们就可以重新训练，学会玩那个特定的游戏。在实验设计情境中，我们经常处于“未知游戏”情境中。这是因为，在我们观察到一些数据之前，几乎不可能知道哪些后续实验将是最佳的。DAD 的方法是模拟不同的可能性，并学会在各个方面都“玩”得好。重新训练的替代方案则是标准贪婪方法和 DAD 之间的混合方法，其中使用一些真实数据在进行过程中重新训练策略。</span></span></p></div><div id="https://www.notion.so/78e00a0baa4246dfa090cf574483e85a" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"></span></p></div></article>
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@@ -187,7 +187,7 @@ <h1 class="Header__Title">深度适应性设计 DAD</h1>
 
 \end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.40006em;vertical-align:-2.45003em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.95003em;"><span style="top:-4.95003em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173142857142857em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.20252142857142857em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.291765em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.2500000000000004em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.45003em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.95003em;"><span style="top:-4.95003em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord"><span class="mord mathbb">E</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.34480000000000005em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="mord mtight">∣</span><span class="mord mtight"><span class="mord mathdefault mtight">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173142857142857em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.20252142857142857em;"><span></span></span></span></span></span></span><span class="mclose mtight">)</span><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.29634285714285713em;"><span style="top:-2.357em;margin-left:-0.03588em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mtight">∣</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.29634285714285713em;"><span style="top:-2.357em;margin-left:-0.04601em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3551999999999999em;"><span></span></span></span></span></span></span><span class="mord"><span class="delimsizing size3">[</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="delimsizing size3">]</span></span></span></span><span style="top:-2.2500000000000004em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord"><span class="mord mathbb">E</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.34480000000000005em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="mord mtight">∣</span><span class="mord mtight"><span class="mord mathdefault mtight">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173142857142857em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.20252142857142857em;"><span></span></span></span></span></span></span><span class="mclose mtight">)</span><span class="mord mathdefault mtight">p</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.29634285714285713em;"><span style="top:-2.357em;margin-left:-0.03588em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mtight">∣</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.29634285714285713em;"><span style="top:-2.357em;margin-left:-0.04601em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3551999999999999em;"><span></span></span></span></span></span></span><span class="mord"><span class="delimsizing size3">[</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="delimsizing size3">]</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.45003em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><div id="https://www.notion.so/26f3c50a2448401c92bee42f7729f6f8" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"></span></p></div></article>
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diff --git a/decision-tree.html b/decision-tree.html
index 6d76167..544ff3e 100644
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@@ -209,7 +209,7 @@ <h1 class="Header__Title">决策树</h1>
 &amp;=\rho\times\Big(\text{Ent}(\tilde{D})-\sum_{v=1}^V\tilde{r}_v\text{Ent}(\tilde{D}^v)\Big)
 \end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.975639em;vertical-align:-2.2378195em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.7378195em;"><span style="top:-5.645965500000001em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord text"><span class="mord">Gain</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">a</span><span class="mclose">)</span></span></span><span style="top:-3.1576295em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.2378195em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.7378195em;"><span style="top:-5.645965500000001em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathdefault">ρ</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord text"><span class="mord">Gain</span></span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201899999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.6023300000000003em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord">~</span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">a</span><span class="mclose">)</span></span></span><span style="top:-3.1576295em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathdefault">ρ</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="delimsizing size2">(</span></span><span class="mord text"><span class="mord">Ent</span></span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201899999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.6023300000000003em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord">~</span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em;">V</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">r</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord">Ent</span></span><span class="mopen">(</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201899999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.6023300000000003em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="delimsizing size2">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.2378195em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><div id="https://www.notion.so/ce5a7c26fd7a420c8ac1f99976ff88cf" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">其中</span></span></p></div><p id="https://www.notion.so/5760c6d7896e4d578f0d25ebd374d45d" class="Equation" data-latex="\text{Ent}(\tilde{D})=-\sum_{k=1}^{|\mathcal{Y}|}\tilde{p}_k\log_2 \tilde{p}_k"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Ent</mtext><mo stretchy="false">(</mo><mover accent="true"><mi>D</mi><mo>~</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="script">Y</mi><mi mathvariant="normal">∣</mi></mrow></munderover><msub><mover accent="true"><mi>p</mi><mo>~</mo></mover><mi>k</mi></msub><msub><mo><mi>log</mi><mo>⁡</mo></mo><mn>2</mn></msub><msub><mover accent="true"><mi>p</mi><mo>~</mo></mover><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\text{Ent}(\tilde{D})=-\sum_{k=1}^{|\mathcal{Y}|}\tilde{p}_k\log_2 \tilde{p}_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1701899999999998em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">Ent</span></span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201899999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.6023300000000003em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord">~</span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.2631180000000004em;vertical-align:-1.302113em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.961005em;"><span style="top:-1.8478869999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.386005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∣</span><span class="mord mtight"><span class="mord mathcal mtight" style="margin-right:0.08222em;">Y</span></span><span class="mord mtight">∣</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.302113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">p</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;"><span class="mord">~</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.20696799999999996em;"><span style="top:-2.4558600000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">p</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;"><span class="mord">~</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p><div id="https://www.notion.so/23c587a734784db38462c0d9a1f4d58f" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">对于</span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">问题2</strong></span><span class="SemanticString">：</span></span></p></div><div id="https://www.notion.so/c1e82c4816954394a232b2968d33a746" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">对于缺失属性值的样本如何将它从父结点划分到子结点中？</span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/31b5a4ebe483484c976e46f2e82581a1" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">若样本</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\boldsymbol{x}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">x</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.44444em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></span></span></span></span></span></span><span class="SemanticString">在划分属性</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span></span><span class="SemanticString">上的取值已知，则将</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\boldsymbol{x}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">x</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.44444em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></span></span></span></span></span></span><span class="SemanticString">划入</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">与其取值对应的子结点</mark></span><span class="SemanticString">，且样本权值在子结点中保持为</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="w_{\boldsymbol{x}}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>w</mi><mi mathvariant="bold-italic">x</mi></msub></mrow><annotation encoding="application/x-tex">w_{\boldsymbol{x}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.161108em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord boldsymbol mtight">x</span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">.</span></span></li><li id="https://www.notion.so/0f425116b7e14f2a8b2470078d289276" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">若样本</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\boldsymbol{x}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">x</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.44444em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></span></span></span></span></span></span><span class="SemanticString">在划分属性</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span></span><span class="SemanticString">上的取值未知，则将</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\boldsymbol{x}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">x</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.44444em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></span></span></span></span></span></span><span class="SemanticString">同时划入所有子结点，且样本权值在子结点中调整为</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\tilde{r}_v\cdot w_{\boldsymbol{x}}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>r</mi><mo>~</mo></mover><mi>v</mi></msub><mo>⋅</mo><msub><mi>w</mi><mi mathvariant="bold-italic">x</mi></msub></mrow><annotation encoding="application/x-tex">\tilde{r}_v\cdot w_{\boldsymbol{x}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8178599999999999em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">r</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.161108em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord boldsymbol mtight">x</span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">，就是</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">让同一个样本以不同的概率划入不同的子结点中</mark></span><span class="SemanticString">。</span></span></li></ul><h2 id="https://www.notion.so/aa02205905d849cc981abae487223323" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--2"><a class="Anchor" href="#https://www.notion.so/aa02205905d849cc981abae487223323"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">多变量决策树</span></span></h2><div id="https://www.notion.so/aa63e0277c874d4b8dc5d305721c908e" class="Image Image--Normal"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F6af50e21-0feb-47af-bdf6-1516cb23de26%2FUntitled.png?width=576&amp;table=block&amp;id=aa63e027-7c87-4d4b-8dc5-d305721c908e"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F6af50e21-0feb-47af-bdf6-1516cb23de26%2FUntitled.png?width=576&amp;table=block&amp;id=aa63e027-7c87-4d4b-8dc5-d305721c908e" style="width:576px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/9551f6ee34164d6398874a88b0cc8c38" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">决策树形成的分类边界的明显特点：轴平行，分类边界由若干个与坐标轴平行的分段组成。</span></span></p></div><div id="https://www.notion.so/5869c969eadb403489adc42664ccf7d8" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">优点：学习结果可解释性强，每个划分都对应一个属性取值</span></span></p></div><div id="https://www.notion.so/b4a305626262484e97dbdf34d7e5b86e" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Ffd2dc29d-5a41-4473-b5ad-a642d3aaf15b%2FUntitled.png?width=1033&amp;table=block&amp;id=b4a30562-6262-484e-97db-df34d7e5b86e"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Ffd2dc29d-5a41-4473-b5ad-a642d3aaf15b%2FUntitled.png?width=1033&amp;table=block&amp;id=b4a30562-6262-484e-97db-df34d7e5b86e" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/50c5d60224ba42d6831939a2d24d9d3c" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">不足：决策树对复杂分类使用分段近似，此时的决策树会相当复杂，由于要进行大量的属性测试，预测时间开销会很大。</span></span></p></div><div id="https://www.notion.so/fbe14c934c3f488187e760b00821399b" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">若能</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">使用斜的划分边界</mark></span><span class="SemanticString">，如图中红色线段所示，则决策树模型将大为简化。“多变量决策树” (multivariate decision tree) 就是能实现这样的“斜划分”甚至更复杂划分的决策树。</span></span></p></div><div id="https://www.notion.so/fc482234ac344077bb8942e6648a530b" class="Image Image--Normal"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F1326ff5d-74fb-4307-8749-1b0ae1f7a815%2FUntitled.png?width=240&amp;table=block&amp;id=fc482234-ac34-4077-bb89-42e6648a530b"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F1326ff5d-74fb-4307-8749-1b0ae1f7a815%2FUntitled.png?width=240&amp;table=block&amp;id=fc482234-ac34-4077-bb89-42e6648a530b" style="width:240px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/b9f4e3e429fc44128a374da2b87daa49" class="Image Image--Normal"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fc2f6f4e6-228e-43d1-a83c-a727740aa6d0%2FUntitled.png?width=624&amp;table=block&amp;id=b9f4e3e4-29fc-4412-8a37-4da2b87daa49"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fc2f6f4e6-228e-43d1-a83c-a727740aa6d0%2FUntitled.png?width=624&amp;table=block&amp;id=b9f4e3e4-29fc-4412-8a37-4da2b87daa49" style="width:624px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/3be485e8c55342dcb848816788b4e320" class="ColumnList"><div id="https://www.notion.so/0f95f8f737354c228eecbcc8580eecd6" class="Column" style="width:calc((100% - var(--column-spacing) * 1) * 0.5)"><div id="https://www.notion.so/24e27daa5aea4ffaba763021421c05e2" class="Image Image--Normal"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fc314ddaa-ba85-4b22-b10b-da047511dd10%2FUntitled.png?width=549&amp;table=block&amp;id=24e27daa-5aea-4ffa-ba76-3021421c05e2"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fc314ddaa-ba85-4b22-b10b-da047511dd10%2FUntitled.png?width=549&amp;table=block&amp;id=24e27daa-5aea-4ffa-ba76-3021421c05e2" style="width:549px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div></div><div id="https://www.notion.so/46364b4e8dc243f6b3b6a0e443111aee" class="Column" style="width:calc((100% - var(--column-spacing) * 1) * 0.5)"><div id="https://www.notion.so/760070a717c14cdc85622b6d695b0283" class="Image Image--Normal"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fe537e962-395a-4541-982a-dd9be8a907fe%2FUntitled.png?width=465&amp;table=block&amp;id=760070a7-17c1-4cdc-8562-2b6d695b0283"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fe537e962-395a-4541-982a-dd9be8a907fe%2FUntitled.png?width=465&amp;table=block&amp;id=760070a7-17c1-4cdc-8562-2b6d695b0283" style="width:465px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div></div></div><h2 id="https://www.notion.so/509ee9f7514e46b5b453277456fe303c" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--2"><a class="Anchor" href="#https://www.notion.so/509ee9f7514e46b5b453277456fe303c"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">总结</span></span></h2><ul class="BulletedListWrapper"><li id="https://www.notion.so/2e72eda600104ae0bd7959a19f103743" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">决策树是一种</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">基于规则的方法</mark></span><span class="SemanticString">，嵌套规则进行预测。根据判断结果进入一个分支，反复执行这个操作直到叶子结点，得到预测结果。这些规则通过训练得到，而非人工制定。</span></span></li><li id="https://www.notion.so/e65646b1bcff4b4695f20b890e7a3b98" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">本质上决策树是通过一系列规则对数据进行分类的过程。首先对数据进行处理，利用归纳算法生成可读的规则和决策树，然后使用决策树对新数据进行分析。</span></span></li><li id="https://www.notion.so/ddc9d94ba56344d4b7bc2a602c94f2be" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">决策树的优点：</span></span><ul class="BulletedListWrapper"><li id="https://www.notion.so/956a39f85a2f45a196d48aea1e019ce3" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">推理过程容易理解，决策推理过程可以表示成 If Then 形式；</span></span></li><li id="https://www.notion.so/4cd276b4ba56467e94aa654b59976be5" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">推理过程完全依赖于属性变量的取值特点；</span></span></li><li id="https://www.notion.so/bd5ac8f622fd4e868362ce9d56d5c698" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">可自动忽略目标变量没有贡献的属性变量，也为判断属性变量的重要性、减少变量的数目提供参考。</span></span></li></ul></li></ul></article>
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diff --git a/index.html b/index.html
index aef03df..295c395 100644
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@@ -869,7 +869,7 @@ <h3 class="Article__Title">
 
 </article>
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diff --git a/kansai.html b/kansai.html
index 4a56546..88234e1 100644
--- a/kansai.html
+++ b/kansai.html
@@ -159,7 +159,7 @@ <h1 class="Header__Title">关西地区随便玩玩</h1>
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   <article id="https://www.notion.so/713ecd3654df4ad29ff3390f7c18ab24" class="PageRoot"><div id="https://www.notion.so/659f34499bd5473ea26d6bdb6b8ff1cb" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fddb6c001-ad85-48cb-9cb6-58a9cc2feae5%2FUntitled.jpeg?width=1279&amp;table=block&amp;id=659f3449-9bd5-473e-a26d-6bdb6b8ff1cb"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fddb6c001-ad85-48cb-9cb6-58a9cc2feae5%2FUntitled.jpeg?width=1279&amp;table=block&amp;id=659f3449-9bd5-473e-a26d-6bdb6b8ff1cb" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/e0bd96dcd1c84226b8b7c20275bbfbc9" class="Image Image--PageWidth"><figure><a 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class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/01dbe4ba3e984bfaac9ed7e89cb11314" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F59234ae5-da4c-4b6e-8934-97d91c8f386a%2FUntitled.jpeg?width=1280&amp;table=block&amp;id=01dbe4ba-3e98-4bfa-ac9e-d7e89cb11314"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F59234ae5-da4c-4b6e-8934-97d91c8f386a%2FUntitled.jpeg?width=1280&amp;table=block&amp;id=01dbe4ba-3e98-4bfa-ac9e-d7e89cb11314" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/f53fa8b8748543a685e54adeb61f9177" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F8ff300d3-fb2b-4921-b71c-f64de6f97ad4%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=f53fa8b8-7485-43a6-85e5-4adeb61f9177"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F8ff300d3-fb2b-4921-b71c-f64de6f97ad4%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=f53fa8b8-7485-43a6-85e5-4adeb61f9177" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/ae7d4db062814dfd916a1bcdd8ba85d2" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fa2e2a2d0-2ccc-49ec-8619-0848dc6aa6fd%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=ae7d4db0-6281-4dfd-916a-1bcdd8ba85d2"><img 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diff --git a/linear-algebra-01.html b/linear-algebra-01.html
index 6705810..80ae6f1 100644
--- a/linear-algebra-01.html
+++ b/linear-algebra-01.html
@@ -226,7 +226,7 @@ <h1 class="Header__Title">[线性代数] 第1章 向量空间</h1>
 U_3&amp;=\{(0,y,y)\in\mathbf{F}^3:y\in\mathbf{F}\}.
 \end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.572324em;vertical-align:-2.036162em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.536162em;"><span style="top:-4.672054em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.147946em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.6238380000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.036162em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.536162em;"><span style="top:-4.672054em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mopen">{</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">F</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathbf">F</span></span><span class="mclose">}</span><span class="mpunct">,</span></span></span><span style="top:-3.147946em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mopen">{</span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">F</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathbf">F</span></span><span class="mclose">}</span><span class="mpunct">,</span></span></span><span style="top:-1.6238380000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mopen">{</span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">F</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathbf">F</span></span><span class="mclose">}</span><span class="mord">.</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.036162em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><div id="https://www.notion.so/28cb86a472254f659e1abd629623a5b5" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">证明 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="U_1+U_2+U_3"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>U</mi><mn>1</mn></msub><mo>+</mo><msub><mi>U</mi><mn>2</mn></msub><mo>+</mo><msub><mi>U</mi><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">U_1+U_2+U_3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 不是直和.</span></span></p></div><div id="https://www.notion.so/a5cf7531ede543dc988f942ae082a931" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">证明</strong></span><span class="SemanticString">  显然 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathbf{F}^3=U_1+U_2+U_3"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="bold">F</mi><mn>3</mn></msup><mo>=</mo><msub><mi>U</mi><mn>1</mn></msub><mo>+</mo><msub><mi>U</mi><mn>2</mn></msub><mo>+</mo><msub><mi>U</mi><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{F}^3=U_1+U_2+U_3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">F</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">，这是因为每个向量 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="(x,y,z)\in\mathbf{F}^3"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi mathvariant="bold">F</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">(x,y,z)\in\mathbf{F}^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">F</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 都可以写成</span></span></p></div><p id="https://www.notion.so/fb713d289a6d42c7b35c723860089f5b" class="Equation" data-latex="(x,y,z)=(x,y,0)+(0,0,z)+(0,0,0),"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">(x,y,z)=(x,y,0)+(0,0,z)+(0,0,0),</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mpunct">,</span></span></span></span></span></p><div id="https://www.notion.so/d3b96f0e193f4a2f8544d7f463da2611" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">右端第一个向量属于 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="U_1"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>U</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">U_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">，第二个向量属于 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="U_2"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>U</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">U_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">，第三个向量属于 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="U_3"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>U</mi><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">U_3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">.</span></span></p></div><div id="https://www.notion.so/857daf21b7cf440d9222a89642f652ec" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">然而 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathbf{F}^3"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="bold">F</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\mathbf{F}^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">F</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 不是 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="U_1,U_2,U_3"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>U</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>U</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>U</mi><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">U_1,U_2,U_3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 的直和，这是因为向量 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="(0,0,0)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,0,0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 能用两种不同方式写成和 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="u_1+u_2+u_3"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><msub><mi>u</mi><mn>2</mn></msub><mo>+</mo><msub><mi>u</mi><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">u_1+u_2+u_3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 使得每个 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="u_j"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">u_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 属于 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="U_j"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>U</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">U_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">.  具体来说，我们有</span></span></p></div><p id="https://www.notion.so/9fc4e695c9de4f0594d915886a964c15" class="Equation" data-latex="(0,0,0)=(0,1,0)+(0,0,1)+(0,-1,-1),"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">(0,0,0)=(0,1,0)+(0,0,1)+(0,-1,-1),</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">,</span></span></span></span></span></p><div id="https://www.notion.so/d1b58d29edf7420cb46a9a07d7b93f14" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">当然也有</span></span></p></div><p id="https://www.notion.so/d8eb5d7e07cc4b5e84b2708773456a59" class="Equation" data-latex="(0,0,0)=(0,0,0)+(0,0,0)+(0,0,0),"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">(0,0,0)=(0,0,0)+(0,0,0)+(0,0,0),</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mpunct">,</span></span></span></span></span></p><div id="https://www.notion.so/cdf353c1b2714cad843a8807c41ec250" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">其中每个等式右端的第一个向量属于 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="U_1"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>U</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">U_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">，第二个向量属于 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="U_2"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>U</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">U_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">，第三个向量属于 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="U_3"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>U</mi><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">U_3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">.</span></span></p></div><div id="https://www.notion.so/ae57d6de1f5a4c1a8e9756ae84cf10c9" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">1.44 直和的条件</strong></span></span></p></div><div id="https://www.notion.so/f3b3d3304d6a4d79909f095c23a6310f" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">设 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="U_1,\dots,U_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>U</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>U</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">U_1,\dots,U_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 都是 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="V"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span></span></span><span class="SemanticString"> 的子空间.  “</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="U_1+\cdots+U_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>U</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>U</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">U_1+\cdots+U_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 是直和”当且仅当“</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span><span class="SemanticString"> 表示成 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="u_1+\cdots+u_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>u</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">u_1+\cdots+u_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">（其中每个 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="u_j\in U_j"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>j</mi></msub><mo>∈</mo><msub><mi>U</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">u_j\in U_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8252079999999999em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">）的唯一方式是每个 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="u_j"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">u_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 都等于 0”.</span></span></p></div><div id="https://www.notion.so/88cd05f581254573a9a0899d2e7ae607" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">证明</strong></span><span class="SemanticString">  首先假设 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="U_1+\cdots+U_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>U</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>U</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">U_1+\cdots+U_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 是直和.  那么直和的定义表明：如果 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="0=u_1+\cdots+u_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>=</mo><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>u</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">0=u_1+\cdots+u_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">（其中每个 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="u_j\in U_j"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>j</mi></msub><mo>∈</mo><msub><mi>U</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">u_j\in U_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8252079999999999em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">），则必有每个 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="u_j"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">u_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 都等于0.</span></span></p></div><div id="https://www.notion.so/3f5badaa30424b309ac3f3d77047b116" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">现在假设：如果 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="0=u_1+\cdots+u_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>=</mo><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>u</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">0=u_1+\cdots+u_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">（其中每个 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="u_j\in U_j"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>j</mi></msub><mo>∈</mo><msub><mi>U</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">u_j\in U_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8252079999999999em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">），则每个 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="u_j"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">u_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 都等于0.  为了证明 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="U_1+\cdots+U_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>U</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>U</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">U_1+\cdots+U_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 是直和，设 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v\in U_1+\cdots+U_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi><mo>∈</mo><msub><mi>U</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>U</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">v\in U_1+\cdots+U_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">.  把 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span></span></span></span></span></span><span class="SemanticString"> 写成</span></span></p></div><p id="https://www.notion.so/d2ddd5023d364069886fa26668469529" class="Equation" data-latex="v=u_1+\cdots+u_m"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi><mo>=</mo><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>u</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">v=u_1+\cdots+u_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p><div id="https://www.notion.so/47183de79fbf43479798d1799a49ec83" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">其中 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="u_1\in U_1,\dots,u_m\in U_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mn>1</mn></msub><mo>∈</mo><msub><mi>U</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>u</mi><mi>m</mi></msub><mo>∈</mo><msub><mi>U</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">u_1\in U_1,\dots,u_m\in U_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">.  为证明这个表示法唯一，假设还有一个表示</span></span></p></div><p id="https://www.notion.so/ec072cb0f5e945fea517f04b012ba53a" class="Equation" data-latex="v=v_1+\cdots+v_m,"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi><mo>=</mo><msub><mi>v</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>v</mi><mi>m</mi></msub><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">v=v_1+\cdots+v_m,</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span></span></p><div id="https://www.notion.so/5fcb1f2d2cf147308726be50364679a8" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">其中 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v_1\in U_1,\dots,v_m\in U_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mn>1</mn></msub><mo>∈</mo><msub><mi>U</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub><mo>∈</mo><msub><mi>U</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">v_1\in U_1,\dots,v_m\in U_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">.  两式相减，我们有</span></span></p></div><p id="https://www.notion.so/c8ec1226bf1d400bb3376e7712041f3d" class="Equation" data-latex="0=(u_1-v_1)+\cdots+(u_m-v_m)."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>=</mo><mo stretchy="false">(</mo><msub><mi>u</mi><mn>1</mn></msub><mo>−</mo><msub><mi>v</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mo>⋯</mo><mo>+</mo><mo stretchy="false">(</mo><msub><mi>u</mi><mi>m</mi></msub><mo>−</mo><msub><mi>v</mi><mi>m</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">0=(u_1-v_1)+\cdots+(u_m-v_m).</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord">.</span></span></span></span></span></p><div id="https://www.notion.so/0c48698188504014a477371e81aa7c33" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">由于 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="u_1-v_1\in U_1,\dots,u_m-v_m\in U_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mn>1</mn></msub><mo>−</mo><msub><mi>v</mi><mn>1</mn></msub><mo>∈</mo><msub><mi>U</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>u</mi><mi>m</mi></msub><mo>−</mo><msub><mi>v</mi><mi>m</mi></msub><mo>∈</mo><msub><mi>U</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">u_1-v_1\in U_1,\dots,u_m-v_m\in U_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">，上式表明每个 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="u_j-v_j"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>j</mi></msub><mo>−</mo><msub><mi>v</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">u_j-v_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8694379999999999em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 都等于 0.  于是，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="u_1=v_1,\dots,u_m=v_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>u</mi><mi>m</mi></msub><mo>=</mo><msub><mi>v</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">u_1=v_1,\dots,u_m=v_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">.</span></span></p></div><div id="https://www.notion.so/6371553b36e04620b57b85d4f105df1d" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">1.45 两个子空间的直和</strong></span></span></p></div><div id="https://www.notion.so/5a1dcf055b164983b721b68e51227157" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">设 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="U"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">U</span></span></span></span></span></span><span class="SemanticString"> 和 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="W"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">W</span></span></span></span></span></span><span class="SemanticString"> 都是 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="V"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span></span></span><span class="SemanticString"> 的子空间，则 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="U+W"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi><mo>+</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">U+W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">W</span></span></span></span></span></span><span class="SemanticString"> 是直和当且仅当 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="U\cap W = \{0\}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi><mo>∩</mo><mi>W</mi><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">U\cap W = \{0\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">W</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mclose">}</span></span></span></span></span></span><span class="SemanticString">.</span></span></p></div><div id="https://www.notion.so/473495ae3dbd480389483275865816f2" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">证明</strong></span><span class="SemanticString">  首先假设 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="U+W"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi><mo>+</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">U+W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">W</span></span></span></span></span></span><span class="SemanticString"> 是直和.  若 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v\in U\cap W"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>U</mi><mo>∩</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">v\in U\cap W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">W</span></span></span></span></span></span><span class="SemanticString">，则 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="0=v+(-v)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>=</mo><mi>v</mi><mo>+</mo><mo stretchy="false">(</mo><mo>−</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">0=v+(-v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">，其中 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v\in U,\, -v\in W"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>U</mi><mo separator="true">,</mo><mtext> </mtext><mo>−</mo><mi>v</mi><mo>∈</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">v\in U,\, -v\in W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">−</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">W</span></span></span></span></span></span><span class="SemanticString">. 由于 0 可唯一地表示成 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="U"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">U</span></span></span></span></span></span><span class="SemanticString"> 中向量与 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="V"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span></span></span><span class="SemanticString"> 中向量的和，我们有 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v=0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">v=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span><span class="SemanticString">.  于是 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="U\cap W = \{0\}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi><mo>∩</mo><mi>W</mi><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">U\cap W = \{0\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">W</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mclose">}</span></span></span></span></span></span><span class="SemanticString">，这就证明了定理的一个方面.</span></span></p></div><div id="https://www.notion.so/5cf6946b8dab49019a46b23b11c4c6fd" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">另一方面，假设 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="U\cap W=\{0\}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi><mo>∩</mo><mi>W</mi><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">U\cap W=\{0\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">W</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mclose">}</span></span></span></span></span></span><span class="SemanticString">.  为证明 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="U+W"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi><mo>+</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">U+W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">W</span></span></span></span></span></span><span class="SemanticString"> 是直和，假设 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="u\in U,\,w\in W,\, 0=u+w"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo>∈</mo><mi>U</mi><mo separator="true">,</mo><mtext> </mtext><mi>w</mi><mo>∈</mo><mi>W</mi><mo separator="true">,</mo><mtext> </mtext><mn>0</mn><mo>=</mo><mi>u</mi><mo>+</mo><mi>w</mi></mrow><annotation encoding="application/x-tex">u\in U,\,w\in W,\, 0=u+w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathdefault">u</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">W</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">u</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span></span></span></span><span class="SemanticString">.  为完成证明，只需证明 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="u=w=0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo>=</mo><mi>w</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">u=w=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">u</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span><span class="SemanticString">（由于 1.44）.  由上面的等式可得 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="u=-w\in W"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo>=</mo><mo>−</mo><mi>w</mi><mo>∈</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">u=-w\in W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">u</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">W</span></span></span></span></span></span><span class="SemanticString">.  于是 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="u\in U\cap W"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo>∈</mo><mi>U</mi><mo>∩</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">u\in U\cap W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathdefault">u</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">W</span></span></span></span></span></span><span class="SemanticString">.  因此 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="u=0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">u=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">u</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span><span class="SemanticString">，由此及上面的等式可得 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="w=0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">w=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span><span class="SemanticString">，这就完成了证明.</span></span></p></div><blockquote id="https://www.notion.so/b67ffe1ee4374d99af12f48a4545b3b7" class="ColorfulBlock ColorfulBlock--ColorDefault Quote"><span class="SemanticStringArray"><span class="SemanticString">子空间的和类似于子集的并.  同样，子空间的直和类似于子集的不交并.  任意两个子空间都相交，因为它们都包含 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span><span class="SemanticString">.  因此要用交为 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\{0\}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{0\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mclose">}</span></span></span></span></span></span><span class="SemanticString"> 代替不相交，至少在两个子空间的情形如此.</span></span></blockquote><div id="https://www.notion.so/ec42fb9ad1904c15b9b085e0c96adea2" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">上面的结果只考虑了两个子空间的情形，在考虑多于两个的子空间的和是否为直和时，只验证任意两个子空间的交为 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\{0\}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{0\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mclose">}</span></span></span></span></span></span><span class="SemanticString"> 是不够的.  为了看出这一点，考虑例 1.43，在那个非直和的例子中，我们有 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="U_1\cap U_2=U_1\cap U_3=U_2\cap U_3=\{0\}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>U</mi><mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi><mn>2</mn></msub><mo>=</mo><msub><mi>U</mi><mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi><mn>3</mn></msub><mo>=</mo><msub><mi>U</mi><mn>2</mn></msub><mo>∩</mo><msub><mi>U</mi><mn>3</mn></msub><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">U_1\cap U_2=U_1\cap U_3=U_2\cap U_3=\{0\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mclose">}</span></span></span></span></span></span><span class="SemanticString">.</span></span></p></div></article>
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diff --git a/linear-algebra-02.html b/linear-algebra-02.html
index b353510..340454b 100644
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@@ -179,7 +179,7 @@ <h1 class="Header__Title">[线性代数] 第2章 有限维向量空间</h1>
 \end{aligned}
 \right.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.500000000000002em;vertical-align:-2.000000000000001em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35002em;"><span style="top:-2.19999em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.19499em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.20499em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.15001em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.2950099999999996em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.30501em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.60002em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8500199999999998em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5000000000000004em;"><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">7</span></span></span><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">4</span></span></span><span style="top:-1.6599999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.000000000000001em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5000000000000004em;"><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span><span style="top:-1.6599999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">−</span><span class="mord">3</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">4</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.000000000000001em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><div id="https://www.notion.so/99d5e1499baf43e88e479bf816f7d4cf" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">无解.</span></span></p></div></li></ul><div id="https://www.notion.so/423322f3222e44c7a575d818e042b587" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">2.5 定义 张成空间（span）</strong></span></span></p></div><div id="https://www.notion.so/8ee1f0a186994435b1973242c23b5120" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="V"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span></span></span><span class="SemanticString"> 中一组向量 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v_1,\dots,v_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">v_1,\dots,v_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 的所有线性组合所构成的集合称为 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v_1,\dots,v_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">v_1,\dots,v_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 的</span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">张成空间</strong></span><span class="SemanticString">，记为</span></span></p></div><div id="https://www.notion.so/a1b4565d072c48fe94e75a050f1f6a13" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"> </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\text{span}(v_1,\dots,v_m)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>span</mtext><mo stretchy="false">(</mo><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{span}(v_1,\dots,v_m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">span</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">.  也就是说，</span></span></p></div><p id="https://www.notion.so/d5969b66f46344fea89dd942644960fb" class="Equation" data-latex="\text{span}(v_1,\dots,v_m)=\{a_1v_1+\cdots+a_mv_m:a_1,\dots,a_m\in\mathbf{F}\}."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>span</mtext><mo stretchy="false">(</mo><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><msub><mi>a</mi><mn>1</mn></msub><msub><mi>v</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>a</mi><mi>m</mi></msub><msub><mi>v</mi><mi>m</mi></msub><mo>:</mo><msub><mi>a</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>a</mi><mi>m</mi></msub><mo>∈</mo><mi mathvariant="bold">F</mi><mo stretchy="false">}</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\text{span}(v_1,\dots,v_m)=\{a_1v_1+\cdots+a_mv_m:a_1,\dots,a_m\in\mathbf{F}\}.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">span</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.7335400000000001em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathbf">F</span></span><span class="mclose">}</span><span class="mord">.</span></span></span></span></span></p><div id="https://www.notion.so/d97657440ff5496d9bd7c6979bb48cff" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">空向量组 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="()"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">()</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 的张成空间定义为 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\{0\}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{0\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mclose">}</span></span></span></span></span></span><span class="SemanticString">.</span></span></p></div><div id="https://www.notion.so/06a825c9382d468ea4b654ef147eb6f6" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">2.6 例</strong></span><span class="SemanticString"> 前面的例子表明在 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathbf{F}^3"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="bold">F</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\mathbf{F}^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">F</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 中，</span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/4e962ef7408f4e0a99b6d247fea7cea6" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="(17,-4,2)\in\text{span}\big((2,1,-3),(1,-2,4)\big)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>17</mn><mo separator="true">,</mo><mo>−</mo><mn>4</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>∈</mo><mtext>span</mtext><mo fence="false">(</mo><mo stretchy="false">(</mo><mn>2</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>2</mn><mo separator="true">,</mo><mn>4</mn><mo stretchy="false">)</mo><mo fence="false">)</mo></mrow><annotation encoding="application/x-tex">(17,-4,2)\in\text{span}\big((2,1,-3),(1,-2,4)\big)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mord">7</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">−</span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em;"></span><span class="mord text"><span class="mord">span</span></span><span class="mord"><span class="delimsizing size1">(</span></span><span class="mopen">(</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">−</span><span class="mord">3</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">−</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">4</span><span class="mclose">)</span><span class="mord"><span class="delimsizing size1">)</span></span></span></span></span></span></span><span class="SemanticString">;</span></span></li><li id="https://www.notion.so/f60fd984326a4a76af7bb247f0ff4f31" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="(17,-4,5)\notin\text{span}\big((2,1,-3),(1,-2,4)\big)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>17</mn><mo separator="true">,</mo><mo>−</mo><mn>4</mn><mo separator="true">,</mo><mn>5</mn><mo stretchy="false">)</mo><mo mathvariant="normal">∉</mo><mtext>span</mtext><mo fence="false">(</mo><mo stretchy="false">(</mo><mn>2</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>2</mn><mo separator="true">,</mo><mn>4</mn><mo stretchy="false">)</mo><mo fence="false">)</mo></mrow><annotation encoding="application/x-tex">(17,-4,5)\notin\text{span}\big((2,1,-3),(1,-2,4)\big)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mord">7</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">−</span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">5</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><span class="mord"><span class="mrel">∈</span></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.75em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="llap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mord">/</span><span class="mspace" style="margin-right:0.05555555555555555em;"></span></span></span><span class="fix"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.25em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em;"></span><span class="mord text"><span class="mord">span</span></span><span class="mord"><span class="delimsizing size1">(</span></span><span class="mopen">(</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">−</span><span class="mord">3</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">−</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">4</span><span class="mclose">)</span><span class="mord"><span class="delimsizing size1">)</span></span></span></span></span></span></span><span class="SemanticString">.</span></span></li></ul><div id="https://www.notion.so/3e1f415aa90d4a0dbaf14d1713c80f02" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">有些数学家采用术语</span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">线性张成空间</strong></span><span class="SemanticString">，意思与张成空间一样.</span></span></p></div><div id="https://www.notion.so/c3cff78c241f411d99a4a1f61ff3a3c9" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">2.7 张成空间是包含这组向量的最小子空间</strong></span></span></p></div><div id="https://www.notion.so/e736db2db5ec4223a52954ae5490e6fc" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="V"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span></span></span><span class="SemanticString"> 中一组向量的张成空间是包含这组向量的最小子空间.</span></span></p></div><div id="https://www.notion.so/82595c631fcf4facbd1918657f05b915" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">证明</strong></span><span class="SemanticString">  设 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v_1,\dots,v_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">v_1,\dots,v_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 是 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="V"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span></span></span><span class="SemanticString"> 中的一组向量.</span></span></p></div><div id="https://www.notion.so/09c79871016349b884615dda8f8f3b9d" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">先证明 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\text{span}(v_1,\dots,v_m)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>span</mtext><mo stretchy="false">(</mo><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{span}(v_1,\dots,v_m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">span</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 是 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="V"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span></span></span><span class="SemanticString"> 的子空间.  加法单位元属于 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\text{span}(v_1,\dots,v_m)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>span</mtext><mo stretchy="false">(</mo><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{span}(v_1,\dots,v_m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">span</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">，因为</span></span></p></div><p id="https://www.notion.so/455b42ebcee64ceb852fbf1a6b51fb2c" class="Equation" data-latex="0=0v_1+\cdots+0v_m."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>=</mo><mn>0</mn><msub><mi>v</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><mn>0</mn><msub><mi>v</mi><mi>m</mi></msub><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">0=0v_1+\cdots+0v_m.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">0</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">0</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span></span></p><div id="https://www.notion.so/0f4641f4d9c14bf3a1757806ce7e2116" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">其次，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\text{span}(v_1,\dots,v_m)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>span</mtext><mo stretchy="false">(</mo><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{span}(v_1,\dots,v_m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">span</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 在加法下封闭，因为</span></span></p></div><p id="https://www.notion.so/affd99d15e1744a2bd25bd039c8980a9" class="Equation" data-latex="(a_1v_1+\cdots+a_mv_m)+(c_1v_1+\cdots+c_mv_m)=(a_1+c_1)v_1+\cdots+(a_m+c_m)v_m."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><msub><mi>v</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>a</mi><mi>m</mi></msub><msub><mi>v</mi><mi>m</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><msub><mi>c</mi><mn>1</mn></msub><msub><mi>v</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>c</mi><mi>m</mi></msub><msub><mi>v</mi><mi>m</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>c</mi><mn>1</mn></msub><mo stretchy="false">)</mo><msub><mi>v</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mi>m</mi></msub><mo>+</mo><msub><mi>c</mi><mi>m</mi></msub><mo stretchy="false">)</mo><msub><mi>v</mi><mi>m</mi></msub><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">(a_1v_1+\cdots+a_mv_m)+(c_1v_1+\cdots+c_mv_m)=(a_1+c_1)v_1+\cdots+(a_m+c_m)v_m.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span></span></p><div id="https://www.notion.so/532d288d31e14331a0e6fc13c8962db5" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">再次，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\text{span}(v_1,\dots,v_m)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>span</mtext><mo stretchy="false">(</mo><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{span}(v_1,\dots,v_m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">span</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 在标量乘法下封闭，因为</span></span></p></div><p id="https://www.notion.so/982124a887f349cdbe19013a030454dc" class="Equation" data-latex="\lambda(a_1v_1+\cdots+a_mv_m)=\lambda a_1v_1+\cdots+\lambda a_mv_m."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><msub><mi>v</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>a</mi><mi>m</mi></msub><msub><mi>v</mi><mi>m</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>λ</mi><msub><mi>a</mi><mn>1</mn></msub><msub><mi>v</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><mi>λ</mi><msub><mi>a</mi><mi>m</mi></msub><msub><mi>v</mi><mi>m</mi></msub><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\lambda(a_1v_1+\cdots+a_mv_m)=\lambda a_1v_1+\cdots+\lambda a_mv_m.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">λ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord mathdefault">λ</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord mathdefault">λ</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span></span></p><div id="https://www.notion.so/fbb866f231b24256b9f0c9d6ddaa9768" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">于是 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\text{span}(v_1,\dots,v_m)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>span</mtext><mo stretchy="false">(</mo><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{span}(v_1,\dots,v_m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">span</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 是 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="V"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span></span></span><span class="SemanticString"> 的子空间（由于 1.34）.</span></span></p></div><div id="https://www.notion.so/8c2a25cbdc0b4b0ba1bf56e703044bdc" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">每个 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v_j"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">v_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 都是 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v_1,\dots,v_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">v_1,\dots,v_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 的线性组合（为了证明这一点，在 2.3 中令 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a_j=1"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">a_j=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span></span><span class="SemanticString"> 并令其他 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span></span><span class="SemanticString"> 都等于 0）.  于是 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\text{span}(v_1,\dots,v_m)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>span</mtext><mo stretchy="false">(</mo><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{span}(v_1,\dots,v_m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">span</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 包含每一个 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v_j"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">v_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">.  反之，由于子空间对加法和标量乘法都封闭，从而 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="V"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span></span></span><span class="SemanticString"> 的包含所有 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v_j"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">v_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 的子空间必定都包含 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\text{span}(v_1,\dots,v_m)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>span</mtext><mo stretchy="false">(</mo><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{span}(v_1,\dots,v_m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">span</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">.  因此 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\text{span}(v_1,\dots,v_m)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>span</mtext><mo stretchy="false">(</mo><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{span}(v_1,\dots,v_m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">span</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 是 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="V"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span></span></span><span class="SemanticString"> 的包含所有向量 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v_1,\dots,v_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">v_1,\dots,v_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 的最小子空间.</span></span></p></div><div id="https://www.notion.so/cb1d892b139248ffa2dc494fec9b55fe" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">2.8 定义 张成（spans）</strong></span></span></p></div><div id="https://www.notion.so/b1145a6e69854d67a5f03da09741bb21" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">若 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\text{span}(v_1,\dots,v_m)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>span</mtext><mo stretchy="false">(</mo><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{span}(v_1,\dots,v_m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">span</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 等于 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="V"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span></span></span><span class="SemanticString">，则称 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v_1,\dots,v_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">v_1,\dots,v_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> </span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">张成</strong></span><span class="SemanticString"> </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="V"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span></span></span><span class="SemanticString">.</span></span></p></div><div id="https://www.notion.so/edc44dc8bafe4f3bba46c31466a4aa23" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">2.9 例</strong></span><span class="SemanticString"> 设 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="n"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span></span></span></span></span></span><span class="SemanticString"> 是正整数.  证明</span></span></p></div><p id="https://www.notion.so/21d821d5e69c40fcad087c836e7e7938" class="Equation" data-latex="(1,0,\dots,0),(0,1,0,\dots,0),\dots,(0,\dots,0,1)"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,0,\dots,0),(0,1,0,\dots,0),\dots,(0,\dots,0,1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></p><div id="https://www.notion.so/c0163897b7dd4249bea7c49267bed2c4" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">张成 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathbf{F}^n"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="bold">F</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{F}^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">F</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">.  上面向量组中的第 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="j"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span></span></span></span></span></span><span class="SemanticString"> 个向量是第 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="j"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span></span></span></span></span></span><span class="SemanticString"> 个元素为 1 其余元素均为 0 的 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="n"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span></span></span></span></span></span><span class="SemanticString"> 元组.</span></span></p></div><div id="https://www.notion.so/9e71617f58944f0c8fceac539f8cb291" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">证明</strong></span><span class="SemanticString">  设 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="(x_1,\dots,x_n)\in\mathbf{F}^n"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><msup><mi mathvariant="bold">F</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">(x_1,\dots,x_n)\in\mathbf{F}^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">F</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">.  则</span></span></p></div><p id="https://www.notion.so/1a8771823d4a45a98afbab6d53efd04d" class="Equation" data-latex="(x_1,\dots,x_n)=x_1(1,0,\dots,0)+x_2(0,1,0,\dots,0)+\cdots+x_n(0,\dots,0,1)."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">(x_1,\dots,x_n)=x_1(1,0,\dots,0)+x_2(0,1,0,\dots,0)+\cdots+x_n(0,\dots,0,1).</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord">.</span></span></span></span></span></p><div id="https://www.notion.so/167daf67b2544e1eb880f97399c33d0a" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">于是 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="(x_1,\dots,x_n)\in\text{span}\big((1,0,\dots,0),(0,1,0,\dots,0),\dots,(0,\dots,0,1)\big)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mtext>span</mtext><mo fence="false">(</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo fence="false">)</mo></mrow><annotation encoding="application/x-tex">(x_1,\dots,x_n)\in\text{span}\big((1,0,\dots,0),(0,1,0,\dots,0),\dots,(0,\dots,0,1)\big)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em;"></span><span class="mord text"><span class="mord">span</span></span><span class="mord"><span class="delimsizing size1">(</span></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord"><span class="delimsizing size1">)</span></span></span></span></span></span></span><span class="SemanticString">.</span></span></p></div><div id="https://www.notion.so/9aa9163e86874f3bbc13ef05d8e3f2a9" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">2.10 定义 有限维向量空间（finite-dimensional vector space）</strong></span></span></p></div><div id="https://www.notion.so/1fc538db362f42c5b3f8850d1d2173be" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">如果一个向量空间可以由该空间的某个向量组张成，则称这个向量空间是</span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">有限维的</strong></span><span class="SemanticString">.</span></span></p></div><blockquote id="https://www.notion.so/3768aef90cc74ba59e45083984240d26" class="ColorfulBlock ColorfulBlock--ColorDefault Quote"><span class="SemanticStringArray"><span class="SemanticString">回想一下，根据定义，每个组都具有有限长度.</span></span></blockquote><div id="https://www.notion.so/2e892be1aa3547808b7d33588c9854a5" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">上面的例 2.9 表明对任意正整数 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="n"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span></span></span></span></span></span><span class="SemanticString">，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathbf{F}^n"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="bold">F</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{F}^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">F</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 是有限维向量空间.</span></span></p></div><div id="https://www.notion.so/ef3de09a772f4e79be6d726aa63dd705" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">2.11 定义 多项式（polynomial），</strong></span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathcal{P}(\mathbf{F})"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">P</mi><mo stretchy="false">(</mo><mi mathvariant="bold">F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{P}(\mathbf{F})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">F</span></span><span class="mclose">)</span></span></span></span></span></strong></span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/b9618d54a53c492894c01cdb2d395b1b" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">对于函数 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p:\mathbf{F}\rightarrow\mathbf{F}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>:</mo><mi mathvariant="bold">F</mi><mo>→</mo><mi mathvariant="bold">F</mi></mrow><annotation encoding="application/x-tex">p:\mathbf{F}\rightarrow\mathbf{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">p</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">F</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">F</span></span></span></span></span></span></span><span class="SemanticString">，若存在 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a_0,\dots,a_m\in\mathbf{F}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>0</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>a</mi><mi>m</mi></msub><mo>∈</mo><mi mathvariant="bold">F</mi></mrow><annotation encoding="application/x-tex">a_0,\dots,a_m\in\mathbf{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335400000000001em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">F</span></span></span></span></span></span></span><span class="SemanticString"> 使得对任意 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="z\in\mathbf{F}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>∈</mo><mi mathvariant="bold">F</mi></mrow><annotation encoding="application/x-tex">z\in\mathbf{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">F</span></span></span></span></span></span></span><span class="SemanticString"> 均有</span></span><p id="https://www.notion.so/f9cf3f3139234c0f91d8404af3011021" class="Equation" data-latex="p(z)=a_0+a_1z+a_2z^2+\cdots+a_mz^m,"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mi>z</mi><mo>+</mo><msub><mi>a</mi><mn>2</mn></msub><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>a</mi><mi>m</mi></msub><msup><mi>z</mi><mi>m</mi></msup><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">p(z)=a_0+a_1z+a_2z^2+\cdots+a_mz^m,</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.0141079999999998em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9088319999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span></span></p><div id="https://www.notion.so/ba9e349188704f3eb317959a45b82dc8" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">则称 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">p</span></span></span></span></span></span><span class="SemanticString"> 为系数属于 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathbf{F}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">F</mi></mrow><annotation encoding="application/x-tex">\mathbf{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">F</span></span></span></span></span></span></span><span class="SemanticString"> 的</span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">多项式</strong></span><span class="SemanticString">.</span></span></p></div></li><li id="https://www.notion.so/74cfe04249034c95bb10311c05638037" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathcal{P}(\mathbf{F})"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">P</mi><mo stretchy="false">(</mo><mi mathvariant="bold">F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{P}(\mathbf{F})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">F</span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 是系数属于 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathbf{F}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">F</mi></mrow><annotation encoding="application/x-tex">\mathbf{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">F</span></span></span></span></span></span></span><span class="SemanticString"> 的全体多项式所组成的集合.</span></span></li></ul><div id="https://www.notion.so/fb5ca2b3717f413eb1e1cf55ccb52386" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">在通常的（多项式）加法和标量乘法下，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathcal{P}(\mathbf{F})"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">P</mi><mo stretchy="false">(</mo><mi mathvariant="bold">F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{P}(\mathbf{F})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">F</span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 是 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathbf{F}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">F</mi></mrow><annotation encoding="application/x-tex">\mathbf{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">F</span></span></span></span></span></span></span><span class="SemanticString"> 上的向量空间.  也就是说，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathcal{P}(\mathbf{F})"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">P</mi><mo stretchy="false">(</mo><mi mathvariant="bold">F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{P}(\mathbf{F})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">F</span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 是 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathbf{F}^\mathbf{F}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="bold">F</mi><mi mathvariant="bold">F</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{F}^\mathbf{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8432769999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">F</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8432769999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">F</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">（</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathbf{F}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">F</mi></mrow><annotation encoding="application/x-tex">\mathbf{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">F</span></span></span></span></span></span></span><span class="SemanticString"> 到 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathbf{F}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">F</mi></mrow><annotation encoding="application/x-tex">\mathbf{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">F</span></span></span></span></span></span></span><span class="SemanticString"> 的全体函数构成的向量空间）的子空间.</span></span></p></div><div id="https://www.notion.so/530cf42944e0468fbedbc90181f2b370" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">一个多项式的系数由该多项式唯一确定.  因此，下面定义的多项式的次数是唯一确定的.</span></span></p></div><div id="https://www.notion.so/7ff8ed05dd344cbf9af9691bee0a7022" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">2.12 定义 多项式的次数（degree of a polynomial），</strong></span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\text{deg}\,p"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>deg</mtext><mtext> </mtext><mi>p</mi></mrow><annotation encoding="application/x-tex">\text{deg}\,p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">deg</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">p</span></span></span></span></span></strong></span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/36841fc23b704baf9278b34ccf27c231" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">对于多项式 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p\in\mathcal{P}(\mathbf{F})"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>∈</mo><mi mathvariant="script">P</mi><mo stretchy="false">(</mo><mi mathvariant="bold">F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p\in\mathcal{P}(\mathbf{F})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335400000000001em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">p</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">F</span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">，若存在标量 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a_0,a_1,\dots,a_m\in\mathbf{F}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>a</mi><mi>m</mi></msub><mo>∈</mo><mi mathvariant="bold">F</mi></mrow><annotation encoding="application/x-tex">a_0,a_1,\dots,a_m\in\mathbf{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335400000000001em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">F</span></span></span></span></span></span></span><span class="SemanticString">，其中 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a_m\ne 0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>m</mi></msub><mo mathvariant="normal">≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a_m\ne 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><span class="mrel"><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.69444em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="rlap"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="inner"><span class="mrel"></span></span><span class="fix"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span><span class="SemanticString">，使得对任意 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="z\in\mathbf{F}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>∈</mo><mi mathvariant="bold">F</mi></mrow><annotation encoding="application/x-tex">z\in\mathbf{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">F</span></span></span></span></span></span></span><span class="SemanticString"> 有</span></span><p id="https://www.notion.so/b91f631a4b4447578ab0ea87ac131ff6" class="Equation" data-latex="p(z)=a_0+a_1z+\cdots+a_mz^m,"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mi>z</mi><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>a</mi><mi>m</mi></msub><msup><mi>z</mi><mi>m</mi></msup><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">p(z)=a_0+a_1z+\cdots+a_mz^m,</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9088319999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span></span></p><div id="https://www.notion.so/6db7313381d54a08941651e5ab3b3476" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">则说 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">p</span></span></span></span></span></span><span class="SemanticString"> 的</span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">次数</strong></span><span class="SemanticString">为 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">m</span></span></span></span></span></span><span class="SemanticString">.  若 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">p</span></span></span></span></span></span><span class="SemanticString"> 的次数为 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">m</span></span></span></span></span></span><span class="SemanticString">，则记 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\text{deg}\,p=m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>deg</mtext><mtext> </mtext><mi>p</mi><mo>=</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">\text{deg}\,p=m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">deg</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">p</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">m</span></span></span></span></span></span><span class="SemanticString">.</span></span></p></div></li><li id="https://www.notion.so/a57dcfa570654478bc46e41bdf5c6c2a" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">规定恒等于 0 的多项式的次数为 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="-\infin"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">-\infin</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord">∞</span></span></span></span></span></span><span class="SemanticString">.</span></span></li></ul><div id="https://www.notion.so/cf3b8308c244491e80f52b786dfcf1d8" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">在下面的定义中，我们约定 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="-\infin&lt;m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi mathvariant="normal">∞</mi><mo>&lt;</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">-\infin&lt;m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord">∞</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">m</span></span></span></span></span></span><span class="SemanticString">，这意味着恒等于 0 的多项式属于 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathcal{P}_m(\mathbf{F})"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">P</mi><mi>m</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{P}_m(\mathbf{F})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">F</span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">.</span></span></p></div><div id="https://www.notion.so/21bcf4e36aaf4e3e8259dc28ed939fca" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">2.13 定义 </strong></span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathcal{P}_m(\mathbf{F})"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">P</mi><mi>m</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{P}_m(\mathbf{F})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">F</span></span><span class="mclose">)</span></span></span></span></span></strong></span></span></p></div><div id="https://www.notion.so/1c99a555a9504e109831eca68ac037bd" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">对于非负整数 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">m</span></span></span></span></span></span><span class="SemanticString">，用 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathcal{P}_m(\mathbf{F})"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">P</mi><mi>m</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{P}_m(\mathbf{F})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">F</span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 表示系数在 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathbf{F}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">F</mi></mrow><annotation encoding="application/x-tex">\mathbf{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">F</span></span></span></span></span></span></span><span class="SemanticString"> 中且次数不超过 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">m</span></span></span></span></span></span><span class="SemanticString"> 的所有多项式构成的集合.</span></span></p></div><div id="https://www.notion.so/2555bac860c14066a6397ca983f3e527" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">要验证下面的例子，只需注意到 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathcal{P}_m(\mathbf{F})=\text{span}(1,z,\dots,z^m)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">P</mi><mi>m</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">F</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>span</mtext><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mi>z</mi><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msup><mi>z</mi><mi>m</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{P}_m(\mathbf{F})=\text{span}(1,z,\dots,z^m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">F</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">span</span></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">.  此处我们用 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="z^k"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>z</mi><mi>k</mi></msup></mrow><annotation encoding="application/x-tex">z^k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.849108em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 表示函数.</span></span></p></div><div id="https://www.notion.so/532cb9a9f57d494e938156b815dddf83" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">2.14 例</strong></span><span class="SemanticString"> 对每个非负整数 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">m</span></span></span></span></span></span><span class="SemanticString">，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathcal{P}_m(\mathbf{F})"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">P</mi><mi>m</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{P}_m(\mathbf{F})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">F</span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 是有限维向量空间.</span></span></p></div><div id="https://www.notion.so/7b6ec63c82ad45c5b404166d6bdecbdd" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">2.15 定义 无限维向量空间（infinite-dimensional vector space）</strong></span></span></p></div><div id="https://www.notion.so/71a918cbba2442ecbc5cce44ccd5ab8c" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">一个向量空间如果不是有限维的，则称为</span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">无限维的</strong></span><span class="SemanticString">.</span></span></p></div><div id="https://www.notion.so/5100a44f9d9641cb9de19bf9fe6a4b4a" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">2.16 例</strong></span><span class="SemanticString"> 证明 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathcal{P}(\mathbf{F})"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">P</mi><mo stretchy="false">(</mo><mi mathvariant="bold">F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{P}(\mathbf{F})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">F</span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 是无限维的.</span></span></p></div><div id="https://www.notion.so/bf71a7abda3f442c8e0375354220a7c5" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">证明</strong></span><span class="SemanticString">  考虑 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathcal{P}(\mathbf{F})"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">P</mi><mo stretchy="false">(</mo><mi mathvariant="bold">F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{P}(\mathbf{F})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">F</span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 中任意一组元素.  记 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">m</span></span></span></span></span></span><span class="SemanticString"> 为这组多项式的最高次数.  则这个组的张成空间中的每个多项式的次数最多为 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">m</span></span></span></span></span></span><span class="SemanticString">.  因此 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="z^{m+1}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>z</mi><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">z^{m+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 不属于这个组的张成空间.  从而没有组能够张成 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathcal{P}(\mathbf{F})"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">P</mi><mo stretchy="false">(</mo><mi mathvariant="bold">F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{P}(\mathbf{F})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">F</span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">.  所以 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathcal{P}(\mathbf{F})"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">P</mi><mo stretchy="false">(</mo><mi mathvariant="bold">F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{P}(\mathbf{F})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">F</span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 是无限维的.</span></span></p></div><h3 id="https://www.notion.so/0f0176a369aa47358facfe66192ccd59" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/0f0176a369aa47358facfe66192ccd59"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">线性无关</span></span></h3><div id="https://www.notion.so/14bb16e40e374e2184d35e2f925a80ef" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">设 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v_1,\dots,v_m\in V"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">v_1,\dots,v_m\in V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335400000000001em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span></span></span><span class="SemanticString"> 且 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v\in\text{span}(v_1,\dots,v_m)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi><mo>∈</mo><mtext>span</mtext><mo stretchy="false">(</mo><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v\in\text{span}(v_1,\dots,v_m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">span</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">.  由张成空间的定义，有 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a_1,\dots,a_m\in\mathbf{F}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>a</mi><mi>m</mi></msub><mo>∈</mo><mi mathvariant="bold">F</mi></mrow><annotation encoding="application/x-tex">a_1,\dots,a_m\in\mathbf{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335400000000001em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">F</span></span></span></span></span></span></span><span class="SemanticString"> 使得</span></span></p></div><p id="https://www.notion.so/4599b54d900b4750b1539fd2ab46a42a" class="Equation" data-latex="v=a_1v_1+\cdots+a_mv_m."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub><msub><mi>v</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>a</mi><mi>m</mi></msub><msub><mi>v</mi><mi>m</mi></msub><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">v=a_1v_1+\cdots+a_mv_m.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span></span></p><div id="https://www.notion.so/08a7e25308134feb9d24e28ef0a78529" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">考虑上式中标量选取的唯一性问题.  假设 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="c_1,\dots,c_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>c</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">c_1,\dots,c_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 是另一组标量也使得</span></span></p></div><p id="https://www.notion.so/4ed1068f5021408c828ef34b580521f3" class="Equation" data-latex="v=c_1v_1+\cdots+c_mv_m."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi><mo>=</mo><msub><mi>c</mi><mn>1</mn></msub><msub><mi>v</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>c</mi><mi>m</mi></msub><msub><mi>v</mi><mi>m</mi></msub><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">v=c_1v_1+\cdots+c_mv_m.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span></span></p><div id="https://www.notion.so/01aad60c54cd4ab6866f1299bf83c08e" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">两式相减得</span></span></p></div><p id="https://www.notion.so/d3cb0f504210497c803bf7e2d1a669ff" class="Equation" data-latex="0=(a_1-c_1)v_1+\cdots+(a_m-c_m)v_m."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>−</mo><msub><mi>c</mi><mn>1</mn></msub><mo stretchy="false">)</mo><msub><mi>v</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mi>m</mi></msub><mo>−</mo><msub><mi>c</mi><mi>m</mi></msub><mo stretchy="false">)</mo><msub><mi>v</mi><mi>m</mi></msub><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">0=(a_1-c_1)v_1+\cdots+(a_m-c_m)v_m.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span></span></p><div id="https://www.notion.so/2b5b594165f64c5da9c40fe813134154" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">于是我们把 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span><span class="SemanticString"> 写成了 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v_1,\dots,v_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">v_1,\dots,v_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 的线性组合.  如果 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span><span class="SemanticString"> 只能用显然的方式（每个标量都取零）写成 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v_1,\dots,v_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">v_1,\dots,v_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 的线性组合，则每个 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a_j-c_j"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub><mo>−</mo><msub><mi>c</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">a_j-c_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8694379999999999em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 都等于 0，即每个 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a_j"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">a_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 都等于 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="c_j"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">c_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">（因此标量的取法确实是唯一的）.  这种情况很重要，所以我们给它起一个特殊的名字——线性无关.</span></span></p></div><div id="https://www.notion.so/9b65cf6f56104236b1480e16e6b15063" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">2.17 定义 线性无关（linearly independent）</strong></span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/13b76bc788cd4707bdc5c481587231c6" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="V"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span></span></span><span class="SemanticString"> 中一组向量 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v_1,\dots,v_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">v_1,\dots,v_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 称为</span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">线性无关</strong></span><span class="SemanticString">，如果使得 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a_1v_1+\cdots+a_mv_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>1</mn></msub><msub><mi>v</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>a</mi><mi>m</mi></msub><msub><mi>v</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">a_1v_1+\cdots+a_mv_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 等于 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span><span class="SemanticString"> 的 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a_1,\dots,a_m\in\mathbf{F}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>a</mi><mi>m</mi></msub><mo>∈</mo><mi mathvariant="bold">F</mi></mrow><annotation encoding="application/x-tex">a_1,\dots,a_m\in\mathbf{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335400000000001em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">F</span></span></span></span></span></span></span><span class="SemanticString"> 只有 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a_1=\cdots=a_m=0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mo>⋯</mo><mo>=</mo><msub><mi>a</mi><mi>m</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a_1=\cdots=a_m=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span><span class="SemanticString">.</span></span></li><li id="https://www.notion.so/5c7d22392e524431834e1e6caca9a80b" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">规定空组 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="()"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">()</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 是线性无关的.</span></span></li></ul><div id="https://www.notion.so/ccbd280e61ea4d41b85b333f59b7f879" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">上一段的推导表明，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v_1,\dots,v_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">v_1,\dots,v_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 是线性无关的当且仅当 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\text{span}(v_1,\dots,v_m)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>span</mtext><mo stretchy="false">(</mo><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{span}(v_1,\dots,v_m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">span</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 中每个向量都可以唯一地表示成 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v_1,\dots,v_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">v_1,\dots,v_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 的线性组合.</span></span></p></div><div id="https://www.notion.so/9b871c873e344791a16ec2cacf8355a7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">2.18 例 线性无关组</strong></span></span></p></div><div id="https://www.notion.so/fb30e4c5229947ad8c6585ecdfefbbb6" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">(a) </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="V"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span></span></span><span class="SemanticString"> 中一个向量 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span></span></span></span></span></span><span class="SemanticString"> 构成的向量组 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span></span></span></span></span></span><span class="SemanticString"> 是线性无关的当且仅当 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v\ne0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi><mo mathvariant="normal">≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">v\ne0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><span class="mrel"><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.69444em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="rlap"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="inner"><span class="mrel"></span></span><span class="fix"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span><span class="SemanticString">.</span></span></p></div><div id="https://www.notion.so/c6e396a7314b4e5487e4c8bf429d7b91" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">(b) </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="V"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span></span></span><span class="SemanticString"> 中两个向量构成的向量组线性无关当且仅当每个向量都不能写成另一个向量的标量倍.</span></span></p></div><div id="https://www.notion.so/4f3d36739b7e4e20b4a221c27c166e82" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">(c) </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathbf{F}^4"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="bold">F</mi><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbf{F}^4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">F</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 中的组 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="(1,0,0,0),(0,1,0,0),(0,0,1,0)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,0,0,0),(0,1,0,0),(0,0,1,0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 线性无关.</span></span></p></div><div id="https://www.notion.so/9cb065f056a442b18a7ba6cf441bc0d7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">(d) 对每个非负整数 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">m</span></span></span></span></span></span><span class="SemanticString">，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathcal{P}(\mathbf{F})"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">P</mi><mo stretchy="false">(</mo><mi mathvariant="bold">F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{P}(\mathbf{F})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">P</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">F</span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 中的组 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="1,z,\dots,z^m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true">,</mo><mi>z</mi><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msup><mi>z</mi><mi>m</mi></msup></mrow><annotation encoding="application/x-tex">1,z,\dots,z^m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.858832em;vertical-align:-0.19444em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 线性无关.</span></span></p></div><div id="https://www.notion.so/834c60cf2b91484582674047c75327dd" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">一个线性无关组中去掉一些向量后，余下的向量构成的向量组仍然线性无关.</span></span></p></div><div id="https://www.notion.so/a850c82846904a6d897b885d03f9c610" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">2.19 定义 线性相关（linear dependent）</strong></span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/dd243dcd57f24c28b1b6b4c1760117e2" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="V"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span></span></span><span class="SemanticString"> 中的一组向量如果不是线性无关的，则称为</span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">线性相关</strong></span><span class="SemanticString">.</span></span></li><li id="https://www.notion.so/c58b3d9ba4334dc4b0607c5eca6a8c92" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">也就是说，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="V"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span></span></span><span class="SemanticString"> 中一组向量 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v_1,\dots,v_m"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">v_1,\dots,v_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 线性相关当且仅当存在不全为零的 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a_1,\dots,a_m\in\mathbf{F}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>a</mi><mi>m</mi></msub><mo>∈</mo><mi mathvariant="bold">F</mi></mrow><annotation encoding="application/x-tex">a_1,\dots,a_m\in\mathbf{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335400000000001em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">F</span></span></span></span></span></span></span><span class="SemanticString"> 使得 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a_1v_1+\cdot+a_mv_m=0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>1</mn></msub><msub><mi>v</mi><mn>1</mn></msub><mo>+</mo><mo>⋅</mo><mo>+</mo><msub><mi>a</mi><mi>m</mi></msub><msub><mi>v</mi><mi>m</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a_1v_1+\cdot+a_mv_m=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span><span class="SemanticString">.</span></span></li></ul><div id="https://www.notion.so/93c8dbd2ccd1489c9647a4cb750fd715" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">2.20 例 线性相关组</strong></span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/e1ea4747b35d4802adad09c046abe25d" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathbf{F}^3"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="bold">F</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\mathbf{F}^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">F</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 中的向量组 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="(2,3,1),(1,-1,2),(7,3,8)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>7</mn><mo separator="true">,</mo><mn>3</mn><mo separator="true">,</mo><mn>8</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,3,1),(1,-1,2),(7,3,8)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord">7</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">8</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 线性相关，因为</span></span><p id="https://www.notion.so/4ef03ff3516a4d6197f972c57052690d" class="Equation" data-latex="2(2,3,1)+3(1,-1,2)+(-1)(7,3,8)=(0,0,0)."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo stretchy="false">(</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mn>3</mn><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>7</mn><mo separator="true">,</mo><mn>3</mn><mo separator="true">,</mo><mn>8</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">2(2,3,1)+3(1,-1,2)+(-1)(7,3,8)=(0,0,0).</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mopen">(</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">3</span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord">7</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">8</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mord">.</span></span></span></span></span></p></li><li id="https://www.notion.so/2c834109d90f40719c90dda7c9b07616" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\mathbf{F}^3"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="bold">F</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\mathbf{F}^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">F</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 中的向量组 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="(2,3,1),(1,-1,2),(7,3,c)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>7</mn><mo separator="true">,</mo><mn>3</mn><mo separator="true">,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,3,1),(1,-1,2),(7,3,c)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord">7</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">c</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 线性相关当且仅当 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="c=8"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mo>=</mo><mn>8</mn></mrow><annotation encoding="application/x-tex">c=8</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">8</span></span></span></span></span></span><span class="SemanticString">.</span></span></li><li id="https://www.notion.so/604cafb82b9440a6be91c1080cc617de" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">若 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="V"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span></span></span><span class="SemanticString"> 中的一组向量中的某个向量是其余向量的线性组合，则这个向量组线性相关.（证明：先将这个向量写成其余向量的线性组合，然后将这个向量移到等式的另一端，并乘以 -1.）</span></span></li><li id="https://www.notion.so/e5924e55fed14196bbe6b1bd59037b77" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">包含 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span><span class="SemanticString"> 向量的向量组线性相关.（这是前一条的特殊情形.）</span></span></li></ul><div id="https://www.notion.so/e97e5b87529747a19b3ef6e91a786830" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"></span></p></div></article>
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diff --git a/mapreduce.html b/mapreduce.html
index 8d96461..d53fb3a 100644
--- a/mapreduce.html
+++ b/mapreduce.html
@@ -176,7 +176,7 @@ <h1 class="Header__Title">MapReduce</h1>
   <span class="token function">Emit</span><span class="token punctuation">(</span><span class="token function">AsString</span><span class="token punctuation">(</span>result<span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">;</span></span></span></span></code></pre><div id="https://www.notion.so/3d4cc5cfb2644ab5871d788fe6ca0874" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">In addition, the user writes code to fill in a </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">mapreduce specification</em></span><span class="SemanticString"> object with the names of the input and output files, and optional tuning parameters. The user then invokes the </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">MapReduce</em></span><span class="SemanticString"> function, passing it the specification object.</span></span></p></div><h3 id="https://www.notion.so/3de2ecbab92d4f1b8aa73814605d2a07" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/3de2ecbab92d4f1b8aa73814605d2a07"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">2.2 Types</span></span></h3><div id="https://www.notion.so/825b804ff86d40928802023896e30696" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">Conceptually the map and reduce functions supplied by the user have associated types:</span></span></p></div><pre id="https://www.notion.so/f7a2b4720bf34ab0ace04738e33b44be" class="Code Code--NoWrap"><code><span class="SemanticStringArray"><span class="SemanticString"><span><span class="token function">map</span>     <span class="token punctuation">(</span>k1<span class="token punctuation">,</span> v1<span class="token punctuation">)</span>       <span class="token operator">-></span> <span class="token function">list</span><span class="token punctuation">(</span>k2<span class="token punctuation">,</span> v2<span class="token punctuation">)</span>
 <span class="token function">reduce</span>  <span class="token punctuation">(</span>k2<span class="token punctuation">,</span> <span class="token function">list</span><span class="token punctuation">(</span>v2<span class="token punctuation">)</span><span class="token punctuation">)</span> <span class="token operator">-></span> <span class="token function">list</span><span class="token punctuation">(</span>v2<span class="token punctuation">)</span></span></span></span></code></pre><div id="https://www.notion.so/86b8f6f0d64c408088cbced09a732bae" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">The input keys and values are drawn from a different domain than the output keys and values. The intermediate keys and values are from the same domain as the output keys and values.</span></span></p></div><h3 id="https://www.notion.so/3de0335119c2430a913d94783c2553a6" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/3de0335119c2430a913d94783c2553a6"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">2.3 More Examples</span></span></h3><ul class="BulletedListWrapper"><li id="https://www.notion.so/25c11baf4c424b28a0fa00af0bee92c9" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">Distributed Grep</strong></span></span><div id="https://www.notion.so/98f5408ea10b45f286d3ef22c802ec9c" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">The map function emits a line if it matches a supplied pattern.</span></span></p></div><div id="https://www.notion.so/d1b5a4999a17421bb4ac0b0dca199977" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">The reduce function is an identity function that just copies the supplied intermediate data to the output.</span></span></p></div></li><li id="https://www.notion.so/3945727a2afc4dfbaeb51acb49388194" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">Count of URL Access Frequency</strong></span></span><div id="https://www.notion.so/ea72a37ac6a94317909b7f320246b064" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">The map function processes logs of web page requests and outputs </span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">&lt;URL, 1&gt;</code></span><span class="SemanticString">.</span></span></p></div><div id="https://www.notion.so/a3a9308d4e1f424791c7e1afd4082fca" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">The reduce function adds together all values for the same URL and emits a </span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">&lt;URL, total count&gt;</code></span><span class="SemanticString"> pair.</span></span></p></div></li><li id="https://www.notion.so/2c64f89bc5a24d659a6ac5b491645891" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">Reverse Web-Link Graph</strong></span></span><div id="https://www.notion.so/6d73b28137dc414c98f7761d23f85f8a" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">The map function outputs </span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">&lt;target, source&gt;</code></span><span class="SemanticString"> pairs for each link to a </span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">target</code></span><span class="SemanticString"> URL found in a page named </span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">source</code></span><span class="SemanticString">.</span></span></p></div><div id="https://www.notion.so/0fae65199aa041c385497125f704f3aa" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">The reduce function concatenates the list of all source URLs associated with a given target URL and emits the pair: </span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">&lt;target, list(source)&gt;</code></span><span class="SemanticString">.</span></span></p></div></li><li id="https://www.notion.so/4e36e815ab2a43ec989629b518446622" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">Term-Vector per Host</strong></span></span><div id="https://www.notion.so/81cc552c48504edcab2db833e3db1d9f" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">A term vector summarizes the most important words that occur in a document or a set of documents as a list of </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="⟨word, frequency⟩"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>w</mi><mi>o</mi><mi>r</mi><mi>d</mi><mo separator="true">,</mo><mi>f</mi><mi>r</mi><mi>e</mi><mi>q</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>y</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">⟨word, frequency⟩</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mord mathdefault">d</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mord mathdefault">e</span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="mord mathdefault">u</span><span class="mord mathdefault">e</span><span class="mord mathdefault">n</span><span class="mord mathdefault">c</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mclose">⟩</span></span></span></span></span></span><span class="SemanticString"> pairs.</span></span></p></div><div id="https://www.notion.so/5a8af4e0df574144b1fb13019e8dd2a4" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">The map function emits a </span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">&lt;hostname, term vector&gt;</code></span><span class="SemanticString"> pair for each input document (where the hostname is extracted from the URL of the document).</span></span></p></div><div id="https://www.notion.so/d9fc6ca3a7f847419c4c1deefad1f52b" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">The reduce function is passed all per-document term vectors for a given host. It adds these term vectors together, throwing away infrequent terms, and then emits a final </span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">&lt;hostname, term vector&gt;</code></span><span class="SemanticString"> pair.</span></span></p></div></li><li id="https://www.notion.so/f1bedcd17e2b4804aee4f6ab43258b99" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">Inverted Index</strong></span></span><div id="https://www.notion.so/2c5a9173a1f946a88ae891349ee340ab" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">The map function parses each document, and emits a sequence of </span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">&lt;word, document ID&gt;</code></span><span class="SemanticString"> pairs.</span></span></p></div><div id="https://www.notion.so/5a9742ad2acf4835af8d00a377a68da3" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">The reduce function accepts all pairs for a given word, sorts the corresponding document IDs and emits a </span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">&lt;word, list(document ID)&gt;</code></span><span class="SemanticString"> pair.</span></span></p></div><div id="https://www.notion.so/b205c5eb4b2e4c33a6d8806c2c122b10" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">The set of all output pairs forms a simple inverted index. It is easy to augment this computation to keep track of word position.</span></span></p></div></li><li id="https://www.notion.so/96cc2598eab5438aaa0d4223d5351f02" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">Distributed Sort</strong></span></span><div id="https://www.notion.so/7e55ad0f2d55457fbb056128f4411f25" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">The map function extracts the key from each record, and emits a </span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">&lt;key, record&gt;</code></span><span class="SemanticString"> pair.</span></span></p></div><div id="https://www.notion.so/6a59e3391e7c49589a8cfeb8dfaa9518" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">The reduce function emits all pairs unchanged.</span></span></p></div><div id="https://www.notion.so/89ba3725d7024cd0933ad1de78696f6c" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">This computation depends on the partitioning facilities described in Section 4.1 and the ordering properties described in Section 4.2.</span></span></p></div></li></ul><h2 id="https://www.notion.so/6a3961d672e542979f65b8e115d3d8c1" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--2"><a class="Anchor" href="#https://www.notion.so/6a3961d672e542979f65b8e115d3d8c1"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">3 Implementation</span></span></h2><div id="https://www.notion.so/681f62df8a4441d0ad750a28c3a75c69" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">This section describes an implementation targeted to the computing environment: large clusters of commodity PCs connected together with switched Ethernet.</span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/0f8fe3a0a5df4dc3b2df22bf23c68f93" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">A cluster consists of hundreds of thousands of machines, and therefore machine failures are common.</span></span></li><li id="https://www.notion.so/9ab9fd21199140448ae0f55743ad4ba3" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">A distributed file system (GFS) is used to manage the data stored on the disks attached directly to individual machines. The file system uses replication to provide availability and reliability on top of unreliable hardware.</span></span></li><li id="https://www.notion.so/ec3539c99a794afa92bdf75fba082341" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">Users submit jobs to a scheduling system. Each job consists of a set of tasks, and is mapped by the scheduler to a set of available machines within a cluster.</span></span></li></ul><h3 id="https://www.notion.so/ffd71347b36741a084756a50e61f4bb4" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/ffd71347b36741a084756a50e61f4bb4"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">3.1 Execution Overview</span></span></h3><div id="https://www.notion.so/e3faf1eae93a49a0a011a33b85f23138" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">The </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">Map</em></span><span class="SemanticString"> invocations are distributed across multiple machines by automatically partitioning the input data into a set of </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">M splits</em></span><span class="SemanticString">. The input splits can be processed in parallel by different machines.</span></span></p></div><div id="https://www.notion.so/75081ac5922141beaaa6ed306ac8566c" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">Reduce</em></span><span class="SemanticString"> invocations are distributed by partitioning the intermediate key space into </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">R</em></span><span class="SemanticString"> pieces using a partitioning function (e.g., </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="hash(key) \mod R"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mi>a</mi><mi>s</mi><mi>h</mi><mo stretchy="false">(</mo><mi>k</mi><mi>e</mi><mi>y</mi><mo stretchy="false">)</mo><mspace></mspace><mspace width="0.6666666666666666em"/><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mi>R</mi></mrow><annotation encoding="application/x-tex">hash(key) \mod R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">h</span><span class="mord mathdefault">a</span><span class="mord mathdefault">s</span><span class="mord mathdefault">h</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mord mathdefault">e</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6666666666666666em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">m</span><span class="mord mathrm">o</span><span class="mord mathrm">d</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span></span></span></span></span></span><span class="SemanticString">). The number of partitions (</span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">R</em></span><span class="SemanticString">) and the partitioning function are specified by the user.</span></span></p></div><div id="https://www.notion.so/38025bc81ea341fe8d53b250cec7417d" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F6f0b4f65-582f-4fc2-9e59-e058a924a615%2Fpdfresizer.com-pdf-crop.svg?width=672&amp;table=block&amp;id=38025bc8-1ea3-41fe-8d53-b250cec7417d"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F6f0b4f65-582f-4fc2-9e59-e058a924a615%2Fpdfresizer.com-pdf-crop.svg?width=672&amp;table=block&amp;id=38025bc8-1ea3-41fe-8d53-b250cec7417d" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/c8e44bce772f4e7b874e86f22c3b5943" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">When the user program calls the </span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">MapReduce</code></span><span class="SemanticString"> function, the following sequence of actions occurs:</span></span></p></div><ol class="NumberedListWrapper"><li id="https://www.notion.so/508f5ec8f8c54f14893aad04c1108775" class="NumberedList" value="1"><span class="SemanticStringArray"><span class="SemanticString">The MapReduce library in the user program first </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown">splits the input files into </span></span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">M</em></span></span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown"> pieces of typically 16 megabytes to 64 megabytes (MB) per piece</span></span><span class="SemanticString"> (controllable by the user via an optional parameter). It then starts up many copies of the program on a cluster of machines.</span></span></li><li id="https://www.notion.so/0d2a15ffdd2241acb19a1e37488dddbb" class="NumberedList" value="2"><span class="SemanticStringArray"><span class="SemanticString">One of the copies of the program is special — the master. The rest are workers that are assigned work by the master. There are </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">M</em></span><span class="SemanticString"> map tasks and </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">R</em></span><span class="SemanticString"> reduce tasks to assign. The master picks idle workers and assigns each one a map task or a reduce task.</span></span></li><li id="https://www.notion.so/960c721780e343f2a61ceda1f7535036" class="NumberedList" value="3"><span class="SemanticStringArray"><span class="SemanticString">A worker who is assigned a map task reads the contents of the corresponding input split. It parses key/value pairs out of the input data and passes each pair to the user-defined </span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">Map</code></span><span class="SemanticString"> function. The intermediate key/value pairs produced by the </span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">Map</code></span><span class="SemanticString"> function are buffered in memory.</span></span></li><li id="https://www.notion.so/c6630496c94947f3b96cb9dc95d026b0" class="NumberedList" value="4"><span class="SemanticStringArray"><span class="SemanticString">Periodically, the buffered pairs are written to local disk, </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown">partitioned into </span></span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">R</em></span></span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown"> regions by the partitioning function</span></span><span class="SemanticString">. The locations of these buffered pairs on the local disk are passed back to the master, who is responsible for forwarding these locations to the reduce workers.</span></span></li><li id="https://www.notion.so/83c5cd2b24d64cc999048707fa76d3b3" class="NumberedList" value="5"><span class="SemanticStringArray"><span class="SemanticString">When a reduce worker is notified by the master about these locations, it uses remote procedure calls to read the buffered data from the local disks of the map workers. When a reduce worker has read all intermediate data, it sorts it by the intermediate keys so that all occurrences of the same key are grouped together. The sorting is needed because typically many different keys map to the same reduce task. If the amount of intermediate data is too large to fit in memory, an external sort is used.</span></span></li><li id="https://www.notion.so/7a93110926414d79a1348d9bb58a6cab" class="NumberedList" value="6"><span class="SemanticStringArray"><span class="SemanticString">The reduce worker iterates over the sorted intermediate data and for each unique intermediate key encountered, it passes the key and the corresponding set of intermediate values to the user’s </span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">Reduce</code></span><span class="SemanticString"> function. The output of the </span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">Reduce</code></span><span class="SemanticString"> function is appended to a final output file for this reduce partition.</span></span></li><li id="https://www.notion.so/c319e46b3d8e434c898faf5113ff5fe6" class="NumberedList" value="7"><span class="SemanticStringArray"><span class="SemanticString">When all map tasks and reduce tasks have been completed, the master wakes up the user program. At this point, the </span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">MapReduce</code></span><span class="SemanticString"> call in the user program returns back to the user code.</span></span></li></ol><div id="https://www.notion.so/abc8ef36c2ca45bd8cc42a6993e61687" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">After successful completion, the output of the mapreduce execution is available in the </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">R</em></span><span class="SemanticString"> output files (one per reduce task, with file names as specified by the user). Typically, users do not need to combine these </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">R</em></span><span class="SemanticString"> output files into one file — they often pass these files as input to another MapReduce call, or use them from another distributed application that is able to deal with input that is partitioned into multiple files.</span></span></p></div><h3 id="https://www.notion.so/0005e16fd3dd403a95cccec17001a332" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/0005e16fd3dd403a95cccec17001a332"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">3.2 Master Data Structures</span></span></h3><div id="https://www.notion.so/8d7a6fc5d9b641e281336f284b94f9b0" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">For each map task and reduce task, the master stores </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown">the state (</span></span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">idle</em></span></span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown">, </span></span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">in-progress</em></span></span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown">, or </span></span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">completed</em></span></span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown">)</span></span><span class="SemanticString">, and </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown">the identity of the worker machine (for non-idle tasks)</span></span><span class="SemanticString">.</span></span></p></div><div id="https://www.notion.so/ef0b9be82ff64a54960b447cc8f8d8aa" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">The master is the conduit through which the location of intermediate file regions is propagated from map tasks to reduce tasks. Therefore, for each completed map task, the master stores </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown">the locations and sizes of the </span></span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">R</em></span></span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown"> intermediate file regions produced by the map task</span></span><span class="SemanticString">. Updates to this location and size information are received as map tasks are completed. The information is pushed incrementally to workers that have </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">in-progress</em></span><span class="SemanticString"> reduce tasks.</span></span></p></div><h3 id="https://www.notion.so/aecd2b296e74458a976af89054407ae9" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/aecd2b296e74458a976af89054407ae9"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">3.3 Fault Tolerance</span></span></h3><div id="https://www.notion.so/f389d7e9fcbe4d1e999258ff2607ce06" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">Worker Failure</strong></span></span></p></div><div id="https://www.notion.so/83f45aae2f6a473aad2b3461ea361bf1" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">The master pings every worker periodically. If no response is received from a worker in a certain amount of time, the master marks the worker as failed.</span></span></p></div><div id="https://www.notion.so/05d40ab2a9b5490bbba85aff6d776351" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown">Completed map tasks</span></span><span class="SemanticString"> are re-executed on a failure because their </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown">output is stored on the </span></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorRed"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown">local disk(s)</span></mark></span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown"> of the failed machine</span></span><span class="SemanticString"> and is therefore inaccessible. </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown">Completed reduce tasks</span></span><span class="SemanticString"> do not need to be re-executed since their </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown">output is stored in </span></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorRed"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown">a global file system</span></mark></span><span class="SemanticString">.</span></span></p></div><div><div></div><div></div><div></div></div><div id="https://www.notion.so/aa340d0873c64fd38f8de9b1273ee992" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">When a map task is executed first by worker </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">A</em></span><span class="SemanticString"> and then later executed by worker </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">B</em></span><span class="SemanticString"> (because </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">A</em></span><span class="SemanticString"> failed), all workers executing reduce tasks are notified of the re-execution. Any reduce tasks that has not already read the data from worker </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">A</em></span><span class="SemanticString"> will read the data from worker </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">B</em></span><span class="SemanticString">.</span></span></p></div><div id="https://www.notion.so/54e4e441467d459b9882e277cec591a3" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">Master Failure</strong></span></span></p></div><div id="https://www.notion.so/4194ef6cf77c4a6dad3270713ce09862" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">The master writes periodic checkpoints of the master data structures described above. If the master task dies, a new copy can be started from the last checkpointed state.</span></span></p></div><div id="https://www.notion.so/5492c9221cdc47ae9342aa24977716ee" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">(However, given that there is only a single master, its failure is unlikely; therefore our current implementation aborts the MapReduce computation if the master fails. Clients can check for this condition and retry the MapReduce operation if they desire.)</span></span></p></div><div id="https://www.notion.so/5394dfb0ea424b269f62efb487bd6c3f" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">Semantics in the Presence of Failures</strong></span></span></p></div><div id="https://www.notion.so/1b82b40e32e046469104e420240ab7c3" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">When the user-supplied </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">map</em></span><span class="SemanticString"> and </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">reduce</em></span><span class="SemanticString"> operators are </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown">deterministic</span></span><span class="SemanticString"> functions of their input values, our distributed implementation produces the same output as would have been produced by a non-faulting </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown">sequential execution</span></span><span class="SemanticString"> of the entire program.</span></span></p></div><div id="https://www.notion.so/c9ca5bf6a30345cfb441e09d8c5471a6" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">We rely on </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown">atomic commits of map and reduce task outputs</span></span><span class="SemanticString"> to achieve this property.</span></span></p></div><div id="https://www.notion.so/4dca4cf9230c43c89aa819f955a9c246" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">Each in-progress task writes its output to private temporary files. A reduce task produces one such file, and a map task produces </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">R</em></span><span class="SemanticString"> such files (one per reduce task).</span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/d393e0b0a8a1461692153cbf1e5622e4" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">When a map task completes, the worker sends a message to the master and includes the names of the </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">R</em></span><span class="SemanticString"> temporary files in the message. </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown">If the master receives a completion message for an already completed map task, it ignores the message.</span></span><span class="SemanticString"> Otherwise, it records the name of </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">R</em></span><span class="SemanticString"> files in a master data structure.</span></span></li><li id="https://www.notion.so/4b6304e25f584447b442a748aaa8d8cc" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">When a reduce task completes, the reduce worker atomically renames its temporary output file to the final output file. If the same reduce task is executed on multiple machines, multiple rename calls will be executed for the same final output file. We rely on the </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown">atomic rename operation provided by the underlying file system</span></span><span class="SemanticString"> to guarantee that the final file system state contains just the data produced by one execution of the reduce task.</span></span></li></ul><div id="https://www.notion.so/dcc0c769e06d40608c70ec556a0a7c09" class="Divider"></div><div id="https://www.notion.so/5cf55bc5089b4dcc870fc0434fe8ef7c" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">When the </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">map</em></span><span class="SemanticString"> and/or </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">reduce</em></span><span class="SemanticString"> operators are </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown">non-deterministic</span></span><span class="SemanticString">, we provide weaker but still reasonable semantics. In the presence of non-deterministic operators, the output of a particular reduce task </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="R_1"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>R</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">R_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> is equivalent to the output for </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="R_1"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>R</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">R_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> produced by a sequential execution of the non-deterministic program. However, the output for a different reduce task </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="R_2"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>R</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">R_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> may correspond to the output for </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="R_2"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>R</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">R_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> produced by a different sequential execution of the non-deterministic program.</span></span></p></div><div id="https://www.notion.so/45c75304a54e48f3a9f59dbe5207ab27" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">Consider map task </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="M"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">M</span></span></span></span></span></span><span class="SemanticString"> and reduce tasks </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="R_1"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>R</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">R_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> and </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="R_2"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>R</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">R_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">. Let </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="e(R_i)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi><mo stretchy="false">(</mo><msub><mi>R</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e(R_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">e</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> be the execution of </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="R_i"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>R</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">R_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> that committed (there is exactly one such execution). The weaker semantics arise because </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="e(R_1)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi><mo stretchy="false">(</mo><msub><mi>R</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e(R_1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">e</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> may have read the output produced by one execution of </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="M"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">M</span></span></span></span></span></span><span class="SemanticString"> and </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="e(R_2)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi><mo stretchy="false">(</mo><msub><mi>R</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e(R_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">e</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> may have read the output produced by a different execution of </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="M"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">M</span></span></span></span></span></span><span class="SemanticString">.</span></span></p></div><h3 id="https://www.notion.so/699cd57da6c54f3595e5a3d9a75667eb" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/699cd57da6c54f3595e5a3d9a75667eb"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">3.4 Locality</span></span></h3><div id="https://www.notion.so/285a244b11944f0da752b3dc8f12d574" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">We conserve network bandwidth by taking advantage of the fact that the input data (managed by GFS) is stored on the local disks of the machines that make up our cluster.</span></span></p></div><div id="https://www.notion.so/96ca9f12b0c944a9ad989ee074574583" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">GFS divides each file into 64 MB blocks, and stores several copies of each block (typically 3 copies) on different machines. The MapReduce master takes the location information of the input files into account and </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Unknown">attempts to schedule a map task on a machine that contains a replica of the corresponding input data</span></span><span class="SemanticString">. Failing that, it attempts to schedule a map task near a replica of that task’s input data (e.g., on a worker machine that is on the same network switch as the machine containing the data).</span></span></p></div><h3 id="https://www.notion.so/9e74a32c4dfa4b47a873164bfc6cdfd8" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/9e74a32c4dfa4b47a873164bfc6cdfd8"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">3.5 Task Granularity</span></span></h3><div id="https://www.notion.so/694b7bab45ae48b9a9612bdc0b1564fb" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">We subdivide the map phase into </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">M</em></span><span class="SemanticString"> pieces and the reduce phase into </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">R</em></span><span class="SemanticString"> pieces. Ideally, </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">M</em></span><span class="SemanticString"> and </span><span class="SemanticString"><em class="SemanticString__Fragment SemanticString__Fragment--Italic">R</em></span><span class="SemanticString"> should be much larger than the number of worker machines.</span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/34321eeda14c47f8a3cf7662ebdaa1dc" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">improve dynamic load balancing</span></span></li><li id="https://www.notion.so/42dcf272be7f470997be3fec62e6b48c" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">speed up recovery when a worker fails: completed map tasks can be spread out across all the other worker machines</span></span></li></ul><div id="https://www.notion.so/8acfda21bbba4e42ad988dee5c046507" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">Practical bounds:</span></span></p></div><div id="https://www.notion.so/8edb4ca57ef44400b475db0758afe092" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">The master must take </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="O(M+R)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>M</mi><mo>+</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(M+R)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> scheduling decisions and keeps </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="O(M*R)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>M</mi><mo>∗</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(M*R)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> state in memory.</span></span></p></div></article>
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diff --git a/mysql-01.html b/mysql-01.html
index 158f684..eb12b59 100644
--- a/mysql-01.html
+++ b/mysql-01.html
@@ -167,7 +167,7 @@ <h1 class="Header__Title">[MySQL 01] SQL查询语句执行流程</h1>
 
 ERROR <span class="token number">1142</span> <span class="token punctuation">(</span><span class="token number">42000</span><span class="token punctuation">)</span>: <span class="token keyword">SELECT</span> command denied <span class="token keyword">to</span> <span class="token keyword">user</span> <span class="token string">'b'</span><span class="token variable">@'localhost'</span> <span class="token keyword">for</span> <span class="token keyword">table</span> <span class="token string">'T'</span></span></span></span></code></pre><div id="https://www.notion.so/882a59ed5da5402a9b513845e5f6c7c2" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">如果有权限，就打开表继续执行。打开表的时候，执行器就会根据表的引擎定义，去使用这个引擎提供的接口。</span></span></p></div><div id="https://www.notion.so/cc77bc58c246481eaa53b25fc552bd38" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">比如这个例子中的表T中，ID字段没有索引，那么执行器的执行流程是这样的：</span></span></p></div><ol class="NumberedListWrapper"><li id="https://www.notion.so/ab0b29f7071a4ac7afaf30616abc0fc0" class="NumberedList" value="1"><span class="SemanticStringArray"><span class="SemanticString">调用InnoDB引擎接口取这个表的第一行，判断ID值是不是10，如果不是则跳过，如果是则将这行存在结果集中；</span></span></li><li id="https://www.notion.so/b183e4ad9d744b55af010045ba83e52c" class="NumberedList" value="2"><span class="SemanticStringArray"><span class="SemanticString">调用引擎接口取“下一行”，重复相同的判断逻辑，直到取到这个表的最后一行。</span></span></li><li id="https://www.notion.so/9ad7f5f194c34fa3b1b4db697fc6eece" class="NumberedList" value="3"><span class="SemanticStringArray"><span class="SemanticString">执行器将上述遍历过程中所有满足条件的行组成的记录集作为结果返回给客户端。</span></span></li></ol><div id="https://www.notion.so/0992ab40f0314a1ebac77e80d2fb60d7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">至此，这个语句就执行完成了。</span></span></p></div><div id="https://www.notion.so/097841badc654de3ba0e63cb6faeb9ab" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">对于有索引的表，执行的逻辑也差不多。第一次调用的是“取满足条件的第一行”这个接口，之后循环取“满足条件的下一行”这个接口，这些接口都是引擎中已经定义好的。</span></span></p></div><div id="https://www.notion.so/30b0211ca3ad4893aec0f036ebabbb59" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">可以在数据库的慢查询日志中看到一个</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">rows_examined</code></span><span class="SemanticString">的字段，表示这个语句执行过程中扫描了多少行。这个值就是在执行器每次调用引擎获取数据行的时候累加的。</span></span></p></div><div id="https://www.notion.so/526d6cd6cbd84ff48b98f86fea139258" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">在某些场景下，执行器调用一次，在引擎内部则扫描了多行，因此</span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">引擎扫描行数跟</strong></span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold"><code class="SemanticString__Fragment SemanticString__Fragment--Code">rows_examined</code></strong></span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">并不是完全相同的</strong></span><span class="SemanticString">。</span></span></p></div></article>
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diff --git a/mysql-02.html b/mysql-02.html
index 00157b3..fad131c 100644
--- a/mysql-02.html
+++ b/mysql-02.html
@@ -165,7 +165,7 @@ <h1 class="Header__Title">[MySQL 02] SQL更新语句执行流程</h1>
 但是由于binlog没写完就crash了，这时候binlog里面就没有记录这个语句。因此，之后备份日志的时候，存起来的binlog里面就没有这条语句。
 然后就会发现，如果需要用这个binlog来恢复临时库的话，由于这个语句的binlog丢失，这个临时库就会少了这一次更新，恢复出来的这一行c的值就是0，与原库的值不同。</span></span></li><li id="https://www.notion.so/d074ac8de9dc44e5875af773bfe1299d" class="NumberedList" value="2"><span class="SemanticStringArray"><span class="SemanticString">先写binlog后写redo log。如果在binlog写完之后crash，由于redo log还没写，崩溃恢复以后这个事务无效，所以这一行c的值是0。但是binlog里面已经记录了“把c从0改成1”这个日志。所以，在之后用binlog来恢复的时候就多了一个事务出来，恢复出来的这一行c的值就是1，与原库的值不同。</span></span></li></ol><div id="https://www.notion.so/471a2da64ec54bc7a806e2f571b9c746" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">可以看到，如果不使用“两阶段提交”，那么数据库的状态就有可能和用它的日志恢复出来的库的状态不一致。</span></span></p></div><div id="https://www.notion.so/33efecf4dda2400d8e2d3d5b550bd062" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">不只是误操作后需要用这个过程来恢复数据。当需要扩容的时候，也就是需要再多搭建一些备库来增加系统的读能力的时候，现在常见的做法也是用全量备份加上应用binlog来实现的，这个“不一致”就会导致线上出现主从数据库不一致的情况。</span></span></p></div><div id="https://www.notion.so/828985a46c464e7ba0b50fea3ba5784d" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">简单说，redo log和binlog都可以用于表示事务的提交状态，而两阶段提交就是让这两个状态保持逻辑上的一致。</span></span></p></div><h3 id="https://www.notion.so/90b09ea066534798969b8a6d4b125e60" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/90b09ea066534798969b8a6d4b125e60"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">小结</span></span></h3><div id="https://www.notion.so/6c8ef5b97ff84d76beba4463927ec27d" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">本节介绍了MySQL里面最重要的两个日志，即物理日志redo log和逻辑日志binlog。</span></span></p></div><div id="https://www.notion.so/094731fba9f344b2bdc37a203ead5dbc" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">redo log用于保证crash-safe能力。</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">innodb_flush_log_ar_trx_commit</code></span><span class="SemanticString">这个参数设置成1的时候，表示每次事务的redo log都直接持久化到磁盘。这个参数建议设置成1，这样可以保证MySQL异常重启之后数据不丢失。</span></span></p></div><div id="https://www.notion.so/e8a1b336b83646889428930eeb8c6a6a" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">sync_binlog</code></span><span class="SemanticString">这个参数设置成1的时候，表示每次事务的binlog都持久化到磁盘。这个参数也建议设置成1，这样可以保证MySQL异常重启之后binlog不丢失。</span></span></p></div><div id="https://www.notion.so/45b4f3253be64de8b826151a349f0406" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">还介绍了与MySQL日志系统密切相关的“两阶段提交”。两阶段提交是跨系统维持数据逻辑一致性时常用的一个方案，即使不做数据库内核开发，日常开发中也有可能会用到。</span></span></p></div><div id="https://www.notion.so/a5cc5db427d346b5bfef2bacc62860b3" class="Divider"></div><div id="https://www.notion.so/c08ffe0eaa4c44dfb4e2da570fdedab6" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">思考题：定期全量备份的周期“取决于系统重要性，有的是一天一备，有的是一周一备”。那么在什么场景下，一天一备会比一周一备更有优势呢？或者说，它影响了数据库系统的哪个指标？</span></span></p></div><div id="https://www.notion.so/594cb2d74382495db4630c2caa9031c7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">好处是“最长恢复时间”更短。</span></span></p></div><div id="https://www.notion.so/54f37338cce342cea7306a4f51d09d1d" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">在一天一备的模式里，最坏情况下需要应用一天的binlog。比如，每天0点做一次全量备份，而要恢复出一个到昨天晚上23点的备份。</span></span></p></div><div id="https://www.notion.so/1a971fcaef2e4d318309fcaa8c628c40" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">一周一备最坏情况就要应用一周的binlog了。</span></span></p></div><div id="https://www.notion.so/cd712b97d57b4ff18936abc7c02861df" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">系统的对应指标就是RTO（恢复目标时间）。</span></span></p></div><div id="https://www.notion.so/e70980509b5b43d7b3e108e220fd95ec" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">当然这个是有成本的，因为更频繁全量备份需要消耗更多存储空间，所以这个RTO是成本换来的，就需要根据业务重要性来评估了。</span></span></p></div></article>
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diff --git a/mysql-03.html b/mysql-03.html
index 1894bd3..220d20c 100644
--- a/mysql-03.html
+++ b/mysql-03.html
@@ -174,7 +174,7 @@ <h1 class="Header__Title">[MySQL 03] 事务隔离</h1>
 
 <span class="token operator">+</span><span class="token comment">-----------------------+----------------+</span></span></span></span></code></pre><div id="https://www.notion.so/7e1c1c665ff44bdcb7ede7ddf01f5c48" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">哪个隔离级别都有它自己的使用场景，要根据自己的业务情况来定。那什么时候需要“可重复读”的场景呢？我们来看一个数据校对逻辑的案例。</span></span></p></div><div id="https://www.notion.so/2854eaeadab340728fe3b8f8e99972d7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">假设你在管理一个个人银行账户表。一个表存了每个月月底的余额，一个表存了账单明细。这时候你要做数据校对，也就是判断上个月的余额和当前余额的差额，是否与本月的账单明细一致。你一定希望在校对过程中，即使有用户发生了一笔新的交易，也不影响你的校对结果。</span></span></p></div><div id="https://www.notion.so/86b7d42e27bd4c4c86fb9f3d6b227119" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">这时候使用“可重复读”隔离级别就很方便。事务启动时的视图可以认为是静态的，不受其他事务更新的影响。</span></span></p></div><h3 id="https://www.notion.so/6d958b38b56e49bc8ad8f9f6c122b010" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/6d958b38b56e49bc8ad8f9f6c122b010"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">事务隔离的实现</span></span></h3><div id="https://www.notion.so/fb8f6e77c00c4fe289d20f9b4347b598" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">理解了事务的隔离级别，再来看看事务隔离具体是怎么实现的。这里展开说明“可重复读”。</span></span></p></div><div id="https://www.notion.so/15331f50e402483395716148dd05a736" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">在MySQL中，实际上每条记录在更新的时候都会同时记录一条回滚操作。记录上的最新值，通过回滚操作，都可以得到前一个状态的值。</span></span></p></div><div id="https://www.notion.so/f2c90487cea6433ca0d90c75c9f39b39" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">假设一个值从1被按顺序改成了2、3、4，在回滚日志里面就会有类似下面的记录。</span></span></p></div><div id="https://www.notion.so/3810409a8c5249e5a8c01ad42b322283" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F1774a30d-a32c-456d-b9a1-37b3b30b49f9%2FUntitled.png?width=1142&amp;table=block&amp;id=3810409a-8c52-49e5-a8c0-1ad42b322283"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F1774a30d-a32c-456d-b9a1-37b3b30b49f9%2FUntitled.png?width=1142&amp;table=block&amp;id=3810409a-8c52-49e5-a8c0-1ad42b322283" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/4ef9c9f2472c4330a8fb15284ebc3e46" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">当前值是4，但是在查询这条记录的时候，不同时刻启动的事务会有不同的read-view。如图中看到的，在视图A、B、C里面，这一个记录的值分别是1、2、4，同一条记录在系统中可以存在多个版本，就是数据库的多版本并发控制（MVCC）。对于read-view A，要得到1，就必须将当前值依次执行图中所有的回滚操作得到。</span></span></p></div><div id="https://www.notion.so/2808e52eb68c4716bbc66edb6a2dc247" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">同时会发现，即使现在有另外一个事务正在将4改成5，这个事务跟read-view A、B、C对应的事务是不会冲突的。</span></span></p></div><div id="https://www.notion.so/6e46921676dc4234bb79db52ca38c13d" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">回滚日志不会一直保留，在不需要的时候才删除。也就是说，系统会判断，当没有事务再需要用到这些回滚日志时，回滚日志会被删除。</span></span></p></div><div id="https://www.notion.so/6194ef63d9c249cea1b60cc62c8bbdfb" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">什么时候才不需要了呢？就是当系统里没有比这个回滚日志更早的read-view的时候。</span></span></p></div><div id="https://www.notion.so/90f0ea0243c445bf8a993f5cea4bce83" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">基于上面的说明，我们来讨论一下为什么建议尽量不要使用长事务。</span></span></p></div><div id="https://www.notion.so/65d5427f8e514cc1a09e4289fe844242" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">长事务意味着系统里面会存在很老的事务视图。由于这些事务随时可能访问数据库里面的任何数据，所以这个事务提交之前，数据库里面它可能用到的回滚记录都必须保留，这就会导致大量占用存储空间。</span></span></p></div><div id="https://www.notion.so/d2a08c8805f5412d8ab73b626a6b1dad" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">在MySQL 5.5及以前的版本，回滚日志是跟数据字典一起放在ibdata文件里的，即使长事务最终提交，回滚段被清理，文件也不会变小。有数据只有20GB，而回滚段有200GB的库。最终只好为了清理回滚段，重建整个库。</span></span></p></div><div id="https://www.notion.so/94df43c06cc24f3487b4203cd5b26165" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">除了对回滚段的影响，长事务还占用锁资源，也可能拖垮整个库。</span></span></p></div><h3 id="https://www.notion.so/504f65be2a5f4b8382a982acf1502c69" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/504f65be2a5f4b8382a982acf1502c69"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">事务的启动方式</span></span></h3><div id="https://www.notion.so/33b26179f311429889a76b1832b049b1" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">如前面所述，长事务有这些潜在风险，建议尽量避免。其实很多时候并不是有意使用长事务，通常是由于误用所致。MySQL的事务启动方式有以下几种：</span></span></p></div><ol class="NumberedListWrapper"><li id="https://www.notion.so/90254708fab74a229024152fee17bf0f" class="NumberedList" value="1"><span class="SemanticStringArray"><span class="SemanticString">显式启动事务语句，</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">begin</code></span><span class="SemanticString">或</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">start transaction</code></span><span class="SemanticString">。配套的提交语句是</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">commit</code></span><span class="SemanticString">，回滚语句是</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">rollback</code></span><span class="SemanticString">。</span></span></li><li id="https://www.notion.so/dab9e8c9cd724f74b2cfdf04302fb8fa" class="NumberedList" value="2"><span class="SemanticStringArray"><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">set autocommit=0</code></span><span class="SemanticString">，这个命令会将这个线程的自动提交关掉。意味着如果只执行一个</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">select</code></span><span class="SemanticString">语句，这个事务就启动了，而且并不会自动提交。这个事务持续存在直到主动执行</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">commit</code></span><span class="SemanticString">或</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">rollback</code></span><span class="SemanticString">语句，或者断开连接。</span></span></li></ol><div id="https://www.notion.so/3328323e59f34df2ae681da70029834a" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">有些客户端连接框架会默认连接成功后先执行一个</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">set autocommit=0</code></span><span class="SemanticString">的命令。这就导致接下来的查询都在事务中，如果是长连接，就导致了意外的长事务。</span></span></p></div><div id="https://www.notion.so/2584c793a57d43a8ac4803dd7e55ac09" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">因此，建议总是使用</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">set autocommit=1</code></span><span class="SemanticString">，通过显式语句的方式来启动事务。</span></span></p></div><div id="https://www.notion.so/599d9b976a644dd592d80a22e722e854" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">但是有人会纠结“多一次交互”的问题。对于一个需要频繁使用事务的业务，第二种方式每个事务在开始时都不需要主动执行一次“</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">begin</code></span><span class="SemanticString">”，减少了语句的交互次数。如果有这个顾虑，建议使用</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">commit work and chain</code></span><span class="SemanticString">语法。</span></span></p></div><div id="https://www.notion.so/a2411b7386964c12a282c1f4cf4991e4" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">在</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">autocommit</code></span><span class="SemanticString">为1的情况下，用</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">begin</code></span><span class="SemanticString">显式启动的事务，如果执行</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">commit</code></span><span class="SemanticString">则提交事务。如果执行</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">commit work and chain</code></span><span class="SemanticString">，则是提交事务并自动启动下一个事务，这样也省去了再次执行</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">begin</code></span><span class="SemanticString">语句的开销。同时带来的好处是从程序开发的角度明确地知道每个语句是否处于事务中。</span></span></p></div><div id="https://www.notion.so/2afc8187fa5e411ca9e8da7fb0095c57" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">可以在information_schema库的innodb_trx这个表中查询长事务，比如下面这个语句，用于查找持续时间超过60s的事务。</span></span></p></div><pre id="https://www.notion.so/12627c86a5d54140ab3d4d65fe4161b4" class="Code Code--NoWrap"><code><span class="SemanticStringArray"><span class="SemanticString"><span><span class="token keyword">select</span> <span class="token operator">*</span> <span class="token keyword">from</span> information_schema<span class="token punctuation">.</span>innodb_trx <span class="token keyword">where</span> TIME_TO_SEC<span class="token punctuation">(</span>timediff<span class="token punctuation">(</span><span class="token function">now</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">,</span>trx_started<span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token operator">></span><span class="token number">60</span></span></span></span></code></pre><h3 id="https://www.notion.so/58127c5d1c3448a1b5640888e45bca38" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/58127c5d1c3448a1b5640888e45bca38"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">小结</span></span></h3><div id="https://www.notion.so/21e8c3e66cd143428eb877cbcc168ed6" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">思考题：有什么方案避免长事务出现或者处理这种情况呢？</span></span></p></div><div id="https://www.notion.so/fa8e6fe08e084f9596afd20fb8d7eb84" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">这个问题，我们可以从应用开发端和数据库端来看。</span></span></p></div><div id="https://www.notion.so/6435c27ab383427fbe0f505f9226d26d" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">首先，从应用开发端来看：</span></span></p></div><ol class="NumberedListWrapper"><li id="https://www.notion.so/b985adf51da2484591e1bec9c1bb58cd" class="NumberedList" value="1"><span class="SemanticStringArray"><span class="SemanticString">确认是否使用了</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">set autocommit=0</code></span><span class="SemanticString">。这个确认工作可以在测试环境中开展，把MySQL的general_log开起来，然后随便跑一个业务逻辑，通过general_log的日志来确认。一般框架如果会设置这个值，也就会提供参数来控制行为，你的目标就是把它改成1。</span></span></li><li id="https://www.notion.so/b0d0112999034141a597524e1ad3f726" class="NumberedList" value="2"><span class="SemanticStringArray"><span class="SemanticString">确认是否有不必要的只读事务。有些框架会习惯不管什么语句先用</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">begin</code></span><span class="SemanticString">/</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">commit</code></span><span class="SemanticString">框起来。有些是业务并没有这个需要，但是也把好几个</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">select</code></span><span class="SemanticString">语句放到了事务中。这种只读事务可以去掉。</span></span></li><li id="https://www.notion.so/5c519deffd0b412396505b17af579bc9" class="NumberedList" value="3"><span class="SemanticStringArray"><span class="SemanticString">业务连接数据库的时候，根据业务本身的预估，通过</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">SET MAX_EXECUTION_TIME</code></span><span class="SemanticString">命令，来控制每个语句执行的最长时间，避免单个语句意外执行太长时间。</span></span></li></ol><div id="https://www.notion.so/fa031d7224c54007ad461853924f8c8f" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">其次，从数据库端来看：</span></span></p></div><ol class="NumberedListWrapper"><li id="https://www.notion.so/f9c786fbc2a142b3a47f4027e2af92bc" class="NumberedList" value="1"><span class="SemanticStringArray"><span class="SemanticString">监控information_schema.innodb_trx表，设置长事务阈值，超过就报警或者kill；</span></span></li><li id="https://www.notion.so/5fc44617e663418aa962e7a40da4f23c" class="NumberedList" value="2"><span class="SemanticStringArray"><span class="SemanticString">Percona的pt-kill工具推荐使用；</span></span></li><li id="https://www.notion.so/590d50b6ebbb4a56bd97a18cac748c18" class="NumberedList" value="3"><span class="SemanticStringArray"><span class="SemanticString">在业务功能测试阶段要求输出所有的general_log，分析日志行为提前发现问题；</span></span></li><li id="https://www.notion.so/225f6b8d53df481aa23f7042bc7e1330" class="NumberedList" value="4"><span class="SemanticStringArray"><span class="SemanticString">如果使用的是MySQL 5.6或者更新版本，把</span><span class="SemanticString"><code class="SemanticString__Fragment SemanticString__Fragment--Code">innodb_undo_tablespaces</code></span><span class="SemanticString">设置成2（或更大的值）。如果真的出现大事务导致回滚段过大，这样设置后清理起来更方便。</span></span></li></ol><div id="https://www.notion.so/4fb68d03b7c749d298ae8f6e2b6cfd59" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"></span></p></div></article>
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diff --git a/mysql-04.html b/mysql-04.html
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@@ -163,7 +163,7 @@ <h1 class="Header__Title">[MySQL 04] 索引（上）</h1>
   </header>
   <article id="https://www.notion.so/3e8c0852f6904565b0e398ed122790e9" class="PageRoot"><div id="https://www.notion.so/210ce86e85744318aa73e9e0e46953b8" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">一句话简单来说，</span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">索引的出现其实就是为了提高数据查询的效率，就像书的目录一样。</strong></span></span></p></div><h3 id="https://www.notion.so/295fffdd7473409eab8a008d9e985c18" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/295fffdd7473409eab8a008d9e985c18"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">索引的常见模型</span></span></h3><div id="https://www.notion.so/d50df833855c4a62b7d5089274d6ee4c" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">可以用于提高读写效率的数据结构很多，这里介绍三种常见、也比较简单的数据结构，它们分别是哈希表、有序数组和搜索树。</span></span></p></div><div id="https://www.notion.so/57741f87abe74653ac1b534f6caf5c92" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">哈希表是一种以键-值（key-value）存储数据的结构，我们只要输入待查找的值即key，就可以找到其对应的值即value。哈希的思路很简单，把值放在数组里，用一个哈希函数把key换算成一个确定的位置，然后把value放在数组的这个位置。</span></span></p></div><div id="https://www.notion.so/e947e926f8e34c8388035e607faa8c4d" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">不可避免地，多个key值经过哈希函数的换算，会出现同一个值的情况。处理这种情况的一种方法是，拉出一个链表。</span></span></p></div><div id="https://www.notion.so/d73d2344a83e410ab91c6d5e85e06ff0" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">假设，现在维护着一个身份证信息和姓名的表，需要根据身份证号查找对应的名字，这时对应的哈希索引的示意图如下所示：</span></span></p></div><div id="https://www.notion.so/0c6dd4dfa8bf451f9e920dac4d89e234" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F12e0b7fd-f9fe-402a-b538-b06dabc070f1%2FUntitled.png?width=1142&amp;table=block&amp;id=0c6dd4df-a8bf-451f-9e92-0dac4d89e234"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F12e0b7fd-f9fe-402a-b538-b06dabc070f1%2FUntitled.png?width=1142&amp;table=block&amp;id=0c6dd4df-a8bf-451f-9e92-0dac4d89e234" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/dc1bb87ddb6b48ff942c775214d6753b" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">图中，User2和User4根据身份证号算出来的值都是N，但没关系，后面还跟了一个链表。假设，这时候要查ID_card_n2对应的名字是什么，处理步骤就是：首先，将ID_card_n2通过哈希函数算出N；然后，按顺序遍历，找到User2。</span></span></p></div><div id="https://www.notion.so/19074bce5d524db7a4b0fb17069e43c5" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">需要注意的是，图中四个ID_card_n的值并不是递增的，这样做的好处是增加新的User时速度会很快，只需要往后追加。但缺点是，因为不是有序的，所以哈希索引做区间查询的速度是很慢的。</span></span></p></div><div id="https://www.notion.so/b9d49c301a374e43a7af8fb3f3534ea7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">可以设想下，如果现在要找身份证号在[ID_card_X, ID_card_Y]这个区间的所有用户，就必须全部扫描一遍了。</span></span></p></div><div id="https://www.notion.so/a39b3e4ca4a547a69058251aece4ac0e" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">所以，</span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">哈希表这种结构适用于只有等值查询的场景</strong></span><span class="SemanticString">，比如Memcached及其他一些NoSQL引擎。</span></span></p></div><div id="https://www.notion.so/3906a592a9344dbe8f348f1459001449" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">而</span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">有序数组在等值查询和范围查询场景中的性能就都非常优秀</strong></span><span class="SemanticString">。还是上面这个根据身份证号查名字的例子，如果使用有序数组来实现的话，示意图如下所示：</span></span></p></div><div id="https://www.notion.so/eae819494a284204b2f7ebf11acca27e" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F9a49dfb8-638b-462a-82a1-68525a8ca524%2FUntitled.png?width=1142&amp;table=block&amp;id=eae81949-4a28-4204-b2f7-ebf11acca27e"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F9a49dfb8-638b-462a-82a1-68525a8ca524%2FUntitled.png?width=1142&amp;table=block&amp;id=eae81949-4a28-4204-b2f7-ebf11acca27e" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/d22295f65dc7410ebc5d531b5c750969" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">这里假设身份证号没有重复，这个数组就是按照身份证号递增的顺序保存的。这时候如果要查ID_card_n2对应的名字，用二分法就可以快速得到，这个时间复杂度是O(log(N))。</span></span></p></div><div id="https://www.notion.so/95944d0da57b41ff92698e2c02b6ebd7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">同时很显然，这个索引结构支持范围查询。要查身份证号在[ID_card_X, ID_card_Y]这个区间的User，可以先用二分法找到ID_card_X（如果不存在ID_card_X，就找到大于ID_card_X的第一个User），然后向右遍历，直到查到第一个大于ID_card_Y的身份证号，退出循环。</span></span></p></div><div id="https://www.notion.so/19d6e4065e1b440791ebf7fa58f85af9" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">如果仅仅看查询效率，有序数组就是最好的数据结构了。但是，在需要更新数据的时候就麻烦了，往中间插入一个记录就必须得挪动后面所有的记录，成本太高。</span></span></p></div><div id="https://www.notion.so/b773d08a0615491894e73d177ec6fc58" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">所以，</span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">有序数组索引只适用于静态存储引擎</strong></span><span class="SemanticString">，比如要保存的是2017年某个城市的所有人口信息，这类不会再修改的数据。</span></span></p></div><div id="https://www.notion.so/baa4b73051a5444ca0eb2b2e32f2f77d" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">二叉搜索树也是经典数据结构。还是上面根据身份证号查名字的例子，如果用二叉搜索树来实现的话，示意图如下所示：</span></span></p></div><div id="https://www.notion.so/b3e87d6bc7724bf7b0a16244cafff761" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F1a8eafbf-e368-4576-8dfa-7552cbffdee0%2FUntitled.png?width=1142&amp;table=block&amp;id=b3e87d6b-c772-4bf7-b0a1-6244cafff761"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F1a8eafbf-e368-4576-8dfa-7552cbffdee0%2FUntitled.png?width=1142&amp;table=block&amp;id=b3e87d6b-c772-4bf7-b0a1-6244cafff761" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/036c2a900ea54022a8ace81648360570" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">二叉搜索树的特点是：每个节点的左儿子小于父节点，父节点又小于右儿子。这样如果要查询ID_card_n2的话，按照图中的搜索顺序就是按照UserA → UserC → UserF → User2这个路径得到。这个时间复杂度是O(log(N))。</span></span></p></div><div id="https://www.notion.so/66358b112a264311928bcade2308f445" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">当然为了维持O(log(N))的时间复杂度，就需要保持这棵树是平衡二叉树。为了做这个保证，更新的时间复杂度也是O(log(N))。</span></span></p></div><div id="https://www.notion.so/14ed7042ca3b4c23b20f0d521538ed6c" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">树可以有二叉，也可以有多叉。多叉树就是每个节点有多个儿子，儿子之间的大小保证从左到右递增。二叉树是搜索效率最高的，但是实际上大多数的数据库存储却并不使用二叉树。其原因是，索引不止存在内存中，还要写到磁盘上。</span></span></p></div><div id="https://www.notion.so/0a35ca1d17ce41989a4ebcccff9e22ac" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">可以想象一下一棵100万节点的平衡二叉树，树高20。一次查询可能需要访问20个数据块。在机械硬盘时代，从磁盘随机读一个数据块需要10ms左右的寻址时间。也就是说，对于一个100万行的表，如果使用二叉树来存储，单独访问一个行可能需要20个10ms的时间，这个查询非常慢。</span></span></p></div><div id="https://www.notion.so/bd2d141b48a04167b29fb230c5bce7f9" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"></span></p></div><div id="https://www.notion.so/a3c80a6e0613403584988885f7b7b992" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"></span></p></div></article>
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diff --git a/not-enough-data-01.html b/not-enough-data-01.html
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--- a/not-enough-data-01.html
+++ b/not-enough-data-01.html
@@ -207,7 +207,7 @@ <h1 class="Header__Title">数据不足下的学习 Part 1：半监督学习</h1>
 p(\text{mixup}_\lambda(y\mid\mathbf{x}_i,\mathbf{x}_j))&amp;\approx\lambda p(y\mid\mathbf{x}_i)+(1-\lambda)p(y\mid\mathbf{x}_i)
 \end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000000000000004em;vertical-align:-1.2500000000000002em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7500000000000002em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord text"><span class="mord">mixup</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.24196799999999993em;"><span style="top:-2.4558600000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">λ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathbf">x</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">x</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord"><span class="mord text"><span class="mord">mixup</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.24196799999999993em;"><span style="top:-2.4558600000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">λ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">x</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">x</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2500000000000002em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7500000000000002em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathdefault">λ</span><span class="mord"><span class="mord"><span class="mord mathbf">x</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">λ</span><span class="mclose">)</span><span class="mord"><span class="mord"><span class="mord mathbf">x</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathdefault">λ</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">x</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">λ</span><span class="mclose">)</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">x</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2500000000000002em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><div id="https://www.notion.so/ccd0c2e107b84476978f12dfdd02d5a1" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">其中 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\theta^\prime"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>θ</mi><mo mathvariant="normal">′</mo></msup></mrow><annotation encoding="application/x-tex">\theta^\prime</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 是 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\theta"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span></span></span><span class="SemanticString"> 的滑动平均，是 mean teacher。</span></span></p></div><div id="https://www.notion.so/01caa8acb85f48c2989c86868d1827c0" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F9bc9970b-ec37-49ab-b2ce-d4869a94aef6%2FUntitled.png?width=1999&amp;table=block&amp;id=01caa8ac-b85f-48c2-989c-86868d1827c0"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F9bc9970b-ec37-49ab-b2ce-d4869a94aef6%2FUntitled.png?width=1999&amp;table=block&amp;id=01caa8ac-b85f-48c2-989c-86868d1827c0" style="width:100%"/></a><figcaption><span class="SemanticStringArray"><span class="SemanticString">图6. Interpolation Consistency Training. MixUp 应用于产生更多带有插值标签的插值样本作为学习目标。</span></span></figcaption></figure></div><div id="https://www.notion.so/2bfa22cb7d6d4d15b5e02358ac0d0dae" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">因为两个随机选择的无标签样本属于不同类别的概率很高（例如在 ImageNet 中有 1000 个物体类），在两个随机无标签样本间应用 mixup 的插值很有可能发生在决策边界附近。根据低密度分离假设，决策边界易于被定位在低密度区域。</span></span></p></div><p id="https://www.notion.so/fe0be2ffb1c443909a36937e9f4bfd9b" class="Equation" data-latex="\mathcal{L}_u^{\text{ICT}}=\mathbb{E}_{\mathbf{u}_i,\mathbf{u}_j∼\mathcal{U}}\mathbb{E}_{\lambda∼\text{Beta}(\alpha,\alpha)}D[p_\theta(y\mid\text{mixup}_\lambda(\mathbf{u}_i,\mathbf{u}_j)),{\text{mixup}}_\lambda(p_{\theta^\prime}(y\mid\mathbf{u}_i),p_{\theta^\prime}(y\mid\mathbf{u}_j))]"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi mathvariant="script">L</mi><mi>u</mi><mtext>ICT</mtext></msubsup><mo>=</mo><msub><mi mathvariant="double-struck">E</mi><mrow><msub><mi mathvariant="bold">u</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi mathvariant="bold">u</mi><mi>j</mi></msub><mo>∼</mo><mi mathvariant="script">U</mi></mrow></msub><msub><mi mathvariant="double-struck">E</mi><mrow><mi>λ</mi><mo>∼</mo><mtext>Beta</mtext><mo stretchy="false">(</mo><mi>α</mi><mo separator="true">,</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></msub><mi>D</mi><mo stretchy="false">[</mo><msub><mi>p</mi><mi>θ</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo>∣</mo><msub><mtext>mixup</mtext><mi>λ</mi></msub><mo stretchy="false">(</mo><msub><mi mathvariant="bold">u</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi mathvariant="bold">u</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo separator="true">,</mo><msub><mtext>mixup</mtext><mi>λ</mi></msub><mo stretchy="false">(</mo><msub><mi>p</mi><msup><mi>θ</mi><mo mathvariant="normal">′</mo></msup></msub><mo stretchy="false">(</mo><mi>y</mi><mo>∣</mo><msub><mi mathvariant="bold">u</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><msub><mi>p</mi><msup><mi>θ</mi><mo mathvariant="normal">′</mo></msup></msub><mo stretchy="false">(</mo><mi>y</mi><mo>∣</mo><msub><mi mathvariant="bold">u</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathcal{L}_u^{\text{ICT}}=\mathbb{E}_{\mathbf{u}_i,\mathbf{u}_j∼\mathcal{U}}\mathbb{E}_{\lambda∼\text{Beta}(\alpha,\alpha)}D[p_\theta(y\mid\text{mixup}_\lambda(\mathbf{u}_i,\mathbf{u}_j)),{\text{mixup}}_\lambda(p_{\theta^\prime}(y\mid\mathbf{u}_i),p_{\theta^\prime}(y\mid\mathbf{u}_j))]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138331em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathcal">L</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">u</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">ICT</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1052em;vertical-align:-0.3551999999999999em;"></span><span class="mord"><span class="mord"><span class="mord mathbb">E</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833100000000004em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathbf mtight">u</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mtight"><span class="mord mathbf mtight">u</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span><span class="mrel mtight">∼</span><span class="mord mtight"><span class="mord mathcal mtight" style="margin-right:0.09931em;">U</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.34731999999999996em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord mathbb">E</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.34480000000000005em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">λ</span><span class="mrel mtight">∼</span><span class="mord text mtight"><span class="mord mtight">Beta</span></span><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.0037em;">α</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:0.0037em;">α</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3551999999999999em;"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord text"><span class="mord">mixup</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.24196799999999993em;"><span style="top:-2.4558600000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">λ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathbf">u</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">u</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mord text"><span class="mord">mixup</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.24196799999999993em;"><span style="top:-2.4558600000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">λ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">u</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">u</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose">)</span><span class="mclose">]</span></span></span></span></span></p><div id="https://www.notion.so/a4b4f76271014e2babc0d510db1dee09" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">其中 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\theta^\prime"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>θ</mi><mo mathvariant="normal">′</mo></msup></mrow><annotation encoding="application/x-tex">\theta^\prime</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 是 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\theta"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span></span></span><span class="SemanticString"> 的滑动平均。</span></span></p></div></article>
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diff --git a/recommendation-01.html b/recommendation-01.html
index f2dffae..717ca32 100644
--- a/recommendation-01.html
+++ b/recommendation-01.html
@@ -314,7 +314,7 @@ <h1 class="Header__Title">利用逆倾向分数 (IPS) 降低选择偏差</h1>
 &amp;=\sum_{(u,i):O_{u,i}=1}\Big(w^TX_{u,i}+\beta_i+\gamma_u\Big)-\sum_{u,i}\log\Big(1+e^{w^TX_{u,i}+\beta_i+\gamma_u}\Big)
 \end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:12.779656000000003em;vertical-align:-6.139828000000002em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.6398280000000005em;"><span style="top:-8.639828em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mord mathdefault">L</span><span class="mopen">(</span><span class="mord mathdefault">ϕ</span><span class="mclose">)</span></span></span><span style="top:-5.476051em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">L</span><span class="mopen">(</span><span class="mord mathdefault">ϕ</span><span class="mclose">)</span></span></span><span style="top:-2.3122739999999986em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"></span></span><span style="top:0.5515030000000023em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.139828000000002em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.6398280000000005em;"><span style="top:-8.639828em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8723309999999997em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∏</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.413777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="delimsizing size4">[</span></span><span class="mord"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.683365em;"><span style="top:-2.26943em;"><span class="pstrut" style="height:3.0063649999999997em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.846935em;"><span style="top:-3.0050700000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7740928571428571em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.07847em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.05278em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285719em;"><span style="top:-2.357em;margin-left:-0.05556em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.2363649999999997em;"><span class="pstrut" style="height:3.0063649999999997em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.683365em;"><span class="pstrut" style="height:3.0063649999999997em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.006365em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9190928571428572em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.07847em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.05278em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285719em;"><span style="top:-2.357em;margin-left:-0.05556em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.820265em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="delimsizing size3">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.689008em;"><span style="top:-3.9029000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.03148em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.263065em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.846935em;"><span style="top:-3.0050700000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7740928571428571em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.07847em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.05278em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285719em;"><span style="top:-2.357em;margin-left:-0.05556em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.820265em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="delimsizing size3">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.689008em;"><span style="top:-3.9029000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.03148em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="delimsizing size4">]</span></span></span></span><span style="top:-5.476051em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8723309999999997em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.413777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mord"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.05278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="delimsizing size2">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="delimsizing size2">(</span></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.056365em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9190928571428572em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.07847em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.05278em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285719em;"><span style="top:-2.357em;margin-left:-0.05556em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="delimsizing size2">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="delimsizing size2">(</span></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.056365em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9190928571428572em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.07847em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.05278em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285719em;"><span style="top:-2.357em;margin-left:-0.05556em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="delimsizing size2">)</span></span><span class="mord"><span class="delimsizing size3">]</span></span></span></span><span style="top:-2.3122739999999986em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8723309999999997em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.413777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mord"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.05278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="delimsizing size2">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="delimsizing size2">(</span></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.056365em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9190928571428572em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.07847em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.05278em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285719em;"><span style="top:-2.357em;margin-left:-0.05556em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="delimsizing size2">)</span></span><span class="mord"><span class="delimsizing size3">]</span></span></span></span><span style="top:0.5515030000000023em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.808995em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span><span class="mclose mtight">)</span><span class="mrel mtight">:</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.538325em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.05278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="delimsizing size2">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8723309999999997em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.413777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="delimsizing size2">(</span></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.056365em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9190928571428572em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.07847em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.05278em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285719em;"><span style="top:-2.357em;margin-left:-0.05556em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="delimsizing size2">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.139828000000002em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><div id="https://www.notion.so/1455098dcce64019aaaa58a84b0d9434" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">简化后的整个模型的 log 似然函数为：</span></span></p></div><p id="https://www.notion.so/5d862377ffdd43bfb591f5eb9f0a0a0d" class="Equation" data-latex="\ell(O|X,\phi)=\sum_{(i,u):O_{u,i}=1}[w^TX_{u,i}+\beta_i+\gamma_u]-\sum_{i,u}\log\Big[1+e^{w^TX_{u,i}+\beta_i+\gamma_u}\Big]."><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℓ</mi><mo stretchy="false">(</mo><mi>O</mi><mi mathvariant="normal">∣</mi><mi>X</mi><mo separator="true">,</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo>∑</mo><mrow><mo stretchy="false">(</mo><mi>i</mi><mo separator="true">,</mo><mi>u</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi>O</mi><mrow><mi>u</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub><mo>=</mo><mn>1</mn></mrow></munder><mo stretchy="false">[</mo><msup><mi>w</mi><mi>T</mi></msup><msub><mi>X</mi><mrow><mi>u</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub><mo>+</mo><msub><mi>β</mi><mi>i</mi></msub><mo>+</mo><msub><mi>γ</mi><mi>u</mi></msub><mo stretchy="false">]</mo><mo>−</mo><munder><mo>∑</mo><mrow><mi>i</mi><mo separator="true">,</mo><mi>u</mi></mrow></munder><mi>log</mi><mo>⁡</mo><mo fence="false">[</mo><mn>1</mn><mo>+</mo><msup><mi>e</mi><mrow><msup><mi>w</mi><mi>T</mi></msup><msub><mi>X</mi><mrow><mi>u</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub><mo>+</mo><msub><mi>β</mi><mi>i</mi></msub><mo>+</mo><msub><mi>γ</mi><mi>u</mi></msub></mrow></msup><mo fence="false">]</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\ell(O|X,\phi)=\sum_{(i,u):O_{u,i}=1}[w^TX_{u,i}+\beta_i+\gamma_u]-\sum_{i,u}\log\Big[1+e^{w^TX_{u,i}+\beta_i+\gamma_u}\Big].</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">ℓ</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">ϕ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.58833em;vertical-align:-1.538325em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.808995em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">u</span><span class="mclose mtight">)</span><span class="mrel mtight">:</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.538325em;"><span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.05278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.563777em;vertical-align:-1.413777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8723309999999997em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">u</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.413777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="delimsizing size2">[</span></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em;"></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.056365em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9190928571428572em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.07847em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.05278em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285719em;"><span style="top:-2.357em;margin-left:-0.05556em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="delimsizing size2">]</span></span><span class="mord">.</span></span></span></span></span></p><div id="https://www.notion.so/9e305ed4b51e48c88d009280854a820b" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">对于任意物品 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="i"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span></span></span><span class="SemanticString"> 对偏置项 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\beta_i"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>β</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\beta_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.05278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 的梯度（</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\gamma_u"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mi>u</mi></msub></mrow><annotation encoding="application/x-tex">\gamma_u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> 与之类似）：</span></span></p></div><p id="https://www.notion.so/8dc72575f4334593883b7ebe01ac1fa6" class="Equation" data-latex="\frac{\partial\ell}{\partial\beta_i}=\sum_uO_{u,i}-\sum_uP_{u,i.}"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi mathvariant="normal">ℓ</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>β</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><munder><mo>∑</mo><mi>u</mi></munder><msub><mi>O</mi><mrow><mi>u</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub><mo>−</mo><munder><mo>∑</mo><mi>u</mi></munder><msub><mi>P</mi><mrow><mi>u</mi><mo separator="true">,</mo><mi>i</mi><mi mathvariant="normal">.</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\frac{\partial\ell}{\partial\beta_i}=\sum_uO_{u,i}-\sum_uP_{u,i.}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.25188em;vertical-align:-0.8804400000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.05278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord">ℓ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.3000100000000003em;vertical-align:-1.250005em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8999949999999999em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">u</span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.250005em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.3000100000000003em;vertical-align:-1.250005em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8999949999999999em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">u</span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.250005em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span><span class="mord mtight">.</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></p><div id="https://www.notion.so/92af90fd06314ce7b33b4650a63655c6" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">令梯度为 0 可解得结论。</span></span></p></div><div id="https://www.notion.so/9d01d5e5b87b444e9e1e7c6ecbf653cb" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"></span></p></div></article>
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-  <div>&copy; Patrick’s Blog 2023</div>
+  <div>&copy; Patrick’s Blog 2024</div>
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diff --git a/svm.html b/svm.html
index c83ba5e..6852c9e 100644
--- a/svm.html
+++ b/svm.html
@@ -209,7 +209,7 @@ <h1 class="Header__Title">支持向量机</h1>
 \xi_i\ge 0
 \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">T</span></span></span></span></span></span></span></span></span></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">b</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><div id="https://www.notion.so/49b93a0f48994167acfcc1e01f81f476" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\phi(x)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">是无限维，无法求整体优化</span></span></p></div><div id="https://www.notion.so/fc6898923e5e421db2f40946a4ba5901" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\phi(x_1)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(x_1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">与</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\phi(x_2)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(x_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">两个无限维向量</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">内积</mark></span><span class="SemanticString">：</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">核函数（Kenerl Function）</mark></span></span></p></div><div id="https://www.notion.so/174d84243d93441eb0716ec0403c5e4e" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="K(x_1,x_2)=\phi(x_1)^\text{T}\phi(x_2)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><msup><mo stretchy="false">)</mo><mtext>T</mtext></msup><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(x_1,x_2)=\phi(x_1)^\text{T}\phi(x_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">T</span></span></span></span></span></span></span></span></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></mark></span></span></p></div><div id="https://www.notion.so/e5c0ee75373043348bfb3f02206b21e7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">只要知道一个核函数，不知道无限维映射</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\phi(x_i)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(x_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">的显式表达，仍然可以求解最优</span></span></p></div><div id="https://www.notion.so/cd925663a90f48c882f5371be44b06e7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">为什么可以通过求内积代替显式</mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\phi"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">ϕ</span></span></span></span></span></mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">——对偶问题</mark></span></span></p></div><div id="https://www.notion.so/e687f6d1cc074893bce028c2f643d47d" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">假设知道了</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="K"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span></span></span></span></span></span><span class="SemanticString">，如何求解</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\phi(x_i)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(x_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">，让优化问题可求解？</span></span></p></div><div id="https://www.notion.so/155c361810394c798437983e5a6a8242" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">核函数也是有限制的，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="K"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span></span></span></span></span></span><span class="SemanticString">要满足某种特定条件，才能拆成内积，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="K(x_1,x_2)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(x_1,x_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">能写成</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\phi(x_1)^\text{T}\phi(x_2)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><msup><mo stretchy="false">)</mo><mtext>T</mtext></msup><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(x_1)^\text{T}\phi(x_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">T</span></span></span></span></span></span></span></span></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">的充要条件：</span></span></p></div><div id="https://www.notion.so/bfaa12c447f343259bdfe81d1fd0f07b" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">（1）交换性：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="K(x_1,x_2)=K(x_2,x_1)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>K</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(x_1,x_2)=K(x_2,x_1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/05a0dd7cc8bb4750b3cdbcfe657821e0" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">（2）半正定性：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\forall C_i,x_i\;,i=1,\dots,N"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi><msub><mi>C</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>x</mi><mi>i</mi></msub><mtext>  </mtext><mo separator="true">,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">\forall C_i,x_i\;,i=1,\dots,N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">∀</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">N</span></span></span></span></span></span><span class="SemanticString">，有</span></span></p></div><p id="https://www.notion.so/723ab4750962433999f24daa33ef1ad7" class="Equation" data-latex="\sum_{i=1}^N\sum_{j=1}^NC_iC_jK(x_i,x_j)\ge0"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><msub><mi>C</mi><mi>i</mi></msub><msub><mi>C</mi><mi>j</mi></msub><mi>K</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>x</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\sum_{i=1}^N\sum_{j=1}^NC_iC_jK(x_i,x_j)\ge0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.2421130000000007em;vertical-align:-1.4137769999999998em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000006em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4137769999999998em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></p><div id="https://www.notion.so/c5c6861db08f4f84a859020ef3377413" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F45235644-a656-4ea9-a3b5-1662ecdfd86e%2FUntitled.png?width=1290&amp;table=block&amp;id=c5c6861d-b08f-4f84-a859-020ef3377413"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F45235644-a656-4ea9-a3b5-1662ecdfd86e%2FUntitled.png?width=1290&amp;table=block&amp;id=c5c6861d-b08f-4f84-a859-020ef3377413" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/47bf4273f84c44a6b336f35c1eacae8d" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F540e3c75-0d5c-4ec2-be39-67adf52aefc0%2FUntitled.png?width=922&amp;table=block&amp;id=47bf4273-f84c-44a6-b336-f35c1eacae8d"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F540e3c75-0d5c-4ec2-be39-67adf52aefc0%2FUntitled.png?width=922&amp;table=block&amp;id=47bf4273-f84c-44a6-b336-f35c1eacae8d" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/6a735a9bbb004d9ea0aff45180635d46" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F5ed5e7e9-248e-4b5f-88b3-5992f8b09b02%2FUntitled.png?width=1016&amp;table=block&amp;id=6a735a9b-bb00-4d9e-a0af-f45180635d46"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F5ed5e7e9-248e-4b5f-88b3-5992f8b09b02%2FUntitled.png?width=1016&amp;table=block&amp;id=6a735a9b-bb00-4d9e-a0af-f45180635d46" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/9e1c2b042a944398aa49f46075a1d1c5" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F88150cbf-00fe-4711-8329-a9eaede3b165%2FUntitled.png?width=960&amp;table=block&amp;id=9e1c2b04-2a94-4398-aa49-f46075a1d1c5"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F88150cbf-00fe-4711-8329-a9eaede3b165%2FUntitled.png?width=960&amp;table=block&amp;id=9e1c2b04-2a94-4398-aa49-f46075a1d1c5" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/c126b026e0f241fbbd26d408d5d140e3" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">高斯核</strong></span></span></p></div><div id="https://www.notion.so/d8abfeb0836146fab461ecc8d93c69e7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">无限维的特征变换</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\phi(x)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/019c9b0607074fa6817dcb56160701b8" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="K(x,x&#x27;)=\exp(-(x-x&#x27;)^2)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mi>exp</mi><mo>⁡</mo><mo stretchy="false">(</mo><mo>−</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(x,x&#x27;)=\exp(-(x-x&#x27;)^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.001892em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">exp</span><span class="mopen">(</span><span class="mord">−</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/f5db0103dc394c519e62963445f22c60" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F427fdce0-1971-414c-96d0-de0c1bd13d5f%2FUntitled.png?width=807&amp;table=block&amp;id=f5db0103-dc39-4c51-9e62-963445f22c60"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F427fdce0-1971-414c-96d0-de0c1bd13d5f%2FUntitled.png?width=807&amp;table=block&amp;id=f5db0103-dc39-4c51-9e62-963445f22c60" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/90b18f069ba546f4b3a03da2fc6e9be9" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">一般形式：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="K(x,x&#x27;)=\exp(-\lambda(x-x&#x27;)^2)\;,\lambda&gt;0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mi>exp</mi><mo>⁡</mo><mo stretchy="false">(</mo><mo>−</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo stretchy="false">)</mo><mtext>  </mtext><mo separator="true">,</mo><mi>λ</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">K(x,x&#x27;)=\exp(-\lambda(x-x&#x27;)^2)\;,\lambda&gt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.001892em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">exp</span><span class="mopen">(</span><span class="mord">−</span><span class="mord mathdefault">λ</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">λ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/f92367ee31fa4d32a0d1b42824e4c2ef" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F04e8056f-c842-4fcd-b0c3-51fb7a5a984f%2FUntitled.png?width=944&amp;table=block&amp;id=f92367ee-31fa-4d32-a0d1-b42824e4c2ef"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F04e8056f-c842-4fcd-b0c3-51fb7a5a984f%2FUntitled.png?width=944&amp;table=block&amp;id=f92367ee-31fa-4d32-a0d1-b42824e4c2ef" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><h2 id="https://www.notion.so/1dfc3708698642ee9f75defa9cc514a4" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--2"><a class="Anchor" href="#https://www.notion.so/1dfc3708698642ee9f75defa9cc514a4"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">对偶问题</span></span></h2><h3 id="https://www.notion.so/88d77d2c5e81400d850ae23d57995701" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/88d77d2c5e81400d850ae23d57995701"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">定义</span></span></h3><div id="https://www.notion.so/8791d7263d8348d5b84902a13cc989c7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"></span></p></div></article>
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+  <div>&copy; Patrick’s Blog 2024</div>
   <div>&centerdot;</div>
   <div>Powered by <a href="https://github.com/dragonman225/notablog" target="_blank"
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diff --git a/tag/Linear Algebra.html b/tag/Linear Algebra.html
index c4ed0ae..48219c8 100644
--- a/tag/Linear Algebra.html	
+++ b/tag/Linear Algebra.html	
@@ -201,7 +201,7 @@ <h3 class="Article__Title">
 
 </article>
   <footer class="Footer">
-  <div>&copy; Patrick’s Blog 2023</div>
+  <div>&copy; Patrick’s Blog 2024</div>
   <div>&centerdot;</div>
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diff --git a/tag/ML.html b/tag/ML.html
index a0a0fba..4dff93d 100644
--- a/tag/ML.html
+++ b/tag/ML.html
@@ -253,7 +253,7 @@ <h3 class="Article__Title">
 
 </article>
   <footer class="Footer">
-  <div>&copy; Patrick’s Blog 2023</div>
+  <div>&copy; Patrick’s Blog 2024</div>
   <div>&centerdot;</div>
   <div>Powered by <a href="https://github.com/dragonman225/notablog" target="_blank"
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diff --git a/tag/Math.html b/tag/Math.html
index a1223db..2e27b5c 100644
--- a/tag/Math.html
+++ b/tag/Math.html
@@ -201,7 +201,7 @@ <h3 class="Article__Title">
 
 </article>
   <footer class="Footer">
-  <div>&copy; Patrick’s Blog 2023</div>
+  <div>&copy; Patrick’s Blog 2024</div>
   <div>&centerdot;</div>
   <div>Powered by <a href="https://github.com/dragonman225/notablog" target="_blank"
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diff --git a/tag/RL.html b/tag/RL.html
index 2105f4b..6a07fe2 100644
--- a/tag/RL.html
+++ b/tag/RL.html
@@ -201,7 +201,7 @@ <h3 class="Article__Title">
 
 </article>
   <footer class="Footer">
-  <div>&copy; Patrick’s Blog 2023</div>
+  <div>&copy; Patrick’s Blog 2024</div>
   <div>&centerdot;</div>
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diff --git a/tag/Recommendation.html b/tag/Recommendation.html
index ddbfe84..e2a654b 100644
--- a/tag/Recommendation.html
+++ b/tag/Recommendation.html
@@ -163,7 +163,7 @@ <h3 class="Article__Title">
 
 </article>
   <footer class="Footer">
-  <div>&copy; Patrick’s Blog 2023</div>
+  <div>&copy; Patrick’s Blog 2024</div>
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diff --git a/tag/Robustness.html b/tag/Robustness.html
index 1745313..017f366 100644
--- a/tag/Robustness.html
+++ b/tag/Robustness.html
@@ -193,7 +193,7 @@ <h3 class="Article__Title">
 
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   <footer class="Footer">
-  <div>&copy; Patrick’s Blog 2023</div>
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diff --git "a/tag/\360\237\223\226Note.html" "b/tag/\360\237\223\226Note.html"
index 1759e1f..10dd406 100644
--- "a/tag/\360\237\223\226Note.html"
+++ "b/tag/\360\237\223\226Note.html"
@@ -691,7 +691,7 @@ <h3 class="Article__Title">
 
 </article>
   <footer class="Footer">
-  <div>&copy; Patrick’s Blog 2023</div>
+  <div>&copy; Patrick’s Blog 2024</div>
   <div>&centerdot;</div>
   <div>Powered by <a href="https://github.com/dragonman225/notablog" target="_blank"
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diff --git "a/tag/\360\237\232\236Memory.html" "b/tag/\360\237\232\236Memory.html"
index cd65556..51ac159 100644
--- "a/tag/\360\237\232\236Memory.html"
+++ "b/tag/\360\237\232\236Memory.html"
@@ -289,7 +289,7 @@ <h3 class="Article__Title">
 
 </article>
   <footer class="Footer">
-  <div>&copy; Patrick’s Blog 2023</div>
+  <div>&copy; Patrick’s Blog 2024</div>
   <div>&centerdot;</div>
   <div>Powered by <a href="https://github.com/dragonman225/notablog" target="_blank"
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diff --git "a/tag/\360\237\247\252BOED.html" "b/tag/\360\237\247\252BOED.html"
index 1245da7..07afa79 100644
--- "a/tag/\360\237\247\252BOED.html"
+++ "b/tag/\360\237\247\252BOED.html"
@@ -263,7 +263,7 @@ <h3 class="Article__Title">
 
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   <footer class="Footer">
-  <div>&copy; Patrick’s Blog 2023</div>
+  <div>&copy; Patrick’s Blog 2024</div>
   <div>&centerdot;</div>
   <div>Powered by <a href="https://github.com/dragonman225/notablog" target="_blank"
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diff --git a/tianjin.html b/tianjin.html
index 9a706dc..f0dca56 100644
--- a/tianjin.html
+++ b/tianjin.html
@@ -159,7 +159,7 @@ <h1 class="Header__Title">天津走一天</h1>
   </header>
   <article id="https://www.notion.so/ac12017898c44ad195ee062e56bf5b7e" class="PageRoot"><div id="https://www.notion.so/d7b17b173759441994c616bb9b648475" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">太热啦！🥵</span></span></p></div><div id="https://www.notion.so/fd9fd8a172d94818ae442eb1516d9201" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F2eb6a201-d0c4-4f32-85f1-a7ced0d31db6%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=fd9fd8a1-72d9-4818-ae44-2eb1516d9201"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F2eb6a201-d0c4-4f32-85f1-a7ced0d31db6%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=fd9fd8a1-72d9-4818-ae44-2eb1516d9201" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/0cfbe8abfddb44fb98f70625680813a0" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F5f2a9f19-c5a4-4c36-a978-ef6967ae7646%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=0cfbe8ab-fddb-44fb-98f7-0625680813a0"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F5f2a9f19-c5a4-4c36-a978-ef6967ae7646%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=0cfbe8ab-fddb-44fb-98f7-0625680813a0" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/8021d40707854946ac443eec00839fdb" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F9caed402-ee98-4726-9812-a07cb2c41795%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=8021d407-0785-4946-ac44-3eec00839fdb"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F9caed402-ee98-4726-9812-a07cb2c41795%2FUntitled.jpeg?width=1702&amp;table=block&amp;id=8021d407-0785-4946-ac44-3eec00839fdb" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/8dbec0f2b6e849d0957b5ff2b7a63c81" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"></span></p></div></article>
   <footer class="Footer">
-  <div>&copy; Patrick’s Blog 2023</div>
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