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<h1 class="Header__Title">贝叶斯模型</h1>
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<span class="DateTagBar__Item DateTagBar__Date">Posted on Wed, Mar 30, 2022</span>
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<article id="https://www.notion.so/eb68c4c62845436ea19d6feebd653bdc" class="PageRoot"><ul id="https://www.notion.so/93bee963a60042a78112edb780b1de6d" class="ColorfulBlock ColorfulBlock--ColorGray TableOfContents"><li class="TableOfContents__Item"><a href="#https://www.notion.so/88769992206c4bf9af53b22ecbb886b7"><div style="margin-left:0px"><span class="SemanticStringArray"><span class="SemanticString">贝叶斯决策论</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/9f8712940d1b4a56824ec9953c744eb8"><div style="margin-left:24px"><span class="SemanticStringArray"><span class="SemanticString">基础知识</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/10f22274fda443d69525f53113a673dd"><div style="margin-left:24px"><span class="SemanticStringArray"><span class="SemanticString">判别模型与生成模型</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/b3e5b9a1d1274e47a64a497c675437e8"><div style="margin-left:0px"><span class="SemanticStringArray"><span class="SemanticString">朴素贝叶斯分类器</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/9f7b4d50cc1e41d69f627a507175fe3a"><div style="margin-left:24px"><span class="SemanticStringArray"><span class="SemanticString">条件独立假设</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/2e98563221604449a04fa3ebf3d1dfa4"><div style="margin-left:24px"><span class="SemanticStringArray"><span class="SemanticString">朴素贝叶斯分类器表达式</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/6519844af81848b0af22816ad8bf8cdf"><div style="margin-left:24px"><span class="SemanticStringArray"><span class="SemanticString">拉普拉斯修正</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/b10cfc69c1204468b4c6fca273beb827"><div style="margin-left:0px"><span class="SemanticStringArray"><span class="SemanticString">半朴素贝叶斯分类器</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/f69ca3142d0747d6ab2c321a80377af2"><div style="margin-left:0px"><span class="SemanticStringArray"><span class="SemanticString">贝叶斯网络</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/28e44ec55aa640e080a6fb3dce4c3985"><div style="margin-left:24px"><span class="SemanticStringArray"><span class="SemanticString">贝叶斯网络的结构形式</span></span></div></a></li></ul><h2 id="https://www.notion.so/88769992206c4bf9af53b22ecbb886b7" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--2"><a class="Anchor" href="#https://www.notion.so/88769992206c4bf9af53b22ecbb886b7"><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">贝叶斯决策论</span></span></h2><h3 id="https://www.notion.so/9f8712940d1b4a56824ec9953c744eb8" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/9f8712940d1b4a56824ec9953c744eb8"><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">基础知识</span></span></h3><div id="https://www.notion.so/94f3c00bae1b41e3be442731514a8233" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">条件概率</strong></span><span class="SemanticString">：描述两个有因果关系的随机事件之间的概率关系，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p(b|a)"><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>b</mi><mi mathvariant="normal">∣</mi><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(b|a)</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">b</span><span class="mord">∣</span><span class="mord mathdefault">a</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">定义为在事件</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">发生的前提下事件</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="b"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</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">b</span></span></span></span></span></span><span class="SemanticString">发生的概率。</span></span></p></div><div id="https://www.notion.so/4108964ad04744b0871e352a1287ab9b" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">贝叶斯公式</strong></span><span class="SemanticString">：两个随机事件之间的关系 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p(b|a)=\dfrac{p(a|b)p(b)}{p(a)}"><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>b</mi><mi mathvariant="normal">∣</mi><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mi mathvariant="normal">∣</mi><mi>b</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">p(b|a)=\dfrac{p(a|b)p(b)}{p(a)}</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">b</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:2.363em;vertical-align:-0.936em;"></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">a</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">a</span><span class="mord">∣</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">b</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></span></span></span></span></span></p></div><div id="https://www.notion.so/bb1afe7084c6473c8bb4d14c22e5a57d" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">分类问题</strong></span><span class="SemanticString">：特征向量取值</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">与样本所属类型</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="y"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</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:0.03588em;">y</span></span></span></span></span></span><span class="SemanticString">有因果关系。因为样本属于类型</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="y"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</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:0.03588em;">y</span></span></span></span></span></span><span class="SemanticString">，所以具有特征值</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">.</span></span></p></div><div id="https://www.notion.so/a00251d2b30b4849acb62aa577a63a41" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">分类器</strong></span><span class="SemanticString">：已知样本的特征向量</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">的条件下反推样本所属的类别。根据贝叶斯公式有 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p(y|\boldsymbol{x})=\dfrac{p(\boldsymbol{x}|y)p(y)}{p(\boldsymbol{x})}"><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 mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mi mathvariant="normal">∣</mi><mi>y</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mrow><mi>p</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">p(y|\boldsymbol{x})=\dfrac{p(\boldsymbol{x}|y)p(y)}{p(\boldsymbol{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">p</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</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:2.363em;vertical-align:-0.936em;"></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"><span class="mord boldsymbol">x</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"><span class="mord boldsymbol">x</span></span></span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</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="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></span></span></span></span><span class="SemanticString">.</span></span></p></div><div id="https://www.notion.so/8cde53c8787742be959b8ba1fb745cd7" 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%2F6ae7aed9-b387-4b8f-8092-3d2183dab2a5%2FUntitled.png?width=480&table=block&id=8cde53c8-7877-42be-959b-8ba1fb745cd7"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F6ae7aed9-b387-4b8f-8092-3d2183dab2a5%2FUntitled.png?width=480&table=block&id=8cde53c8-7877-42be-959b-8ba1fb745cd7" style="width:480px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/cded612cb122421db733a263f4bfdab0" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">最小化分类错误率的贝叶斯最优分类器 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="h^*(x)=\underset{c \in \mathcal{Y}}{\argmax}P(c|\boldsymbol{x})"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>h</mi><mo>∗</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi><munder><mo><mi mathvariant="normal">arg max</mi><mo></mo></mo><mrow><mi>c</mi><mo>∈</mo><mi mathvariant="script">Y</mi></mrow></munder></mi><mi>P</mi><mo stretchy="false">(</mo><mi>c</mi><mi mathvariant="normal">∣</mi><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h^*(x)=\underset{c \in \mathcal{Y}}{\argmax}P(c|\boldsymbol{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"><span class="mord mathdefault">h</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="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:1.756825em;vertical-align:-1.006825em;"></span><span class="mord"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.43055999999999983em;"><span style="top:-2.161229em;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">c</span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mathcal mtight" style="margin-right:0.08222em;">Y</span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><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></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.006825em;"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathdefault">c</span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> 对于每个样本</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">，选择使后验概率</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="P(c|\boldsymbol{x})"><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>c</mi><mi mathvariant="normal">∣</mi><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(c|\boldsymbol{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" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathdefault">c</span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">最大的类别标记。（</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="P(c|\boldsymbol{x})"><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>c</mi><mi mathvariant="normal">∣</mi><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(c|\boldsymbol{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" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathdefault">c</span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">现实中难以直接获得）</span></span></p></div><h3 id="https://www.notion.so/10f22274fda443d69525f53113a673dd" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/10f22274fda443d69525f53113a673dd"><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">判别模型与生成模型</span></span></h3><div id="https://www.notion.so/dd452a9f81064f6abd84d7b4e3fb210b" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">机器学习所要实现的是</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">基于有限的训练样本尽可能准确地估计出后验概率</mark></span><span class="SemanticString">。</span></span></p></div><div id="https://www.notion.so/45933d5b54cc49f4906d7a2b32da0e81" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">两种基本策略：</span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/d8d02776f04f4fb887a4ad039da6e15f" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">判别式模型</strong></span></span><ul class="BulletedListWrapper"><li id="https://www.notion.so/8572a8e13ef043e9b4fef8baefe1ba60" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">直接建模</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="P(c|\boldsymbol{x})"><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>c</mi><mi mathvariant="normal">∣</mi><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(c|\boldsymbol{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" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathdefault">c</span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></span><span class="mclose">)</span></span></span></span></span></span></span></li><li id="https://www.notion.so/15603cff6b784628a9f67ffeb9ed4738" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">决策树、BP神经网络、SVM</span></span></li></ul></li><li id="https://www.notion.so/41f84b50f3084f199dc24fde1a8619b1" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">生成式模型</strong></span></span><ul class="BulletedListWrapper"><li id="https://www.notion.so/afd9bc2d37454ce5aad5a4bb20c65091" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">先建模联合概率分布</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="P(\boldsymbol{x}, 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 mathvariant="bold-italic">x</mi><mo separator="true">,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(\boldsymbol{x}, 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" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></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">，再计算</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="P(c|\boldsymbol{x})"><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>c</mi><mi mathvariant="normal">∣</mi><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(c|\boldsymbol{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" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathdefault">c</span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></span><span class="mclose">)</span></span></span></span></span></span></span><div id="https://www.notion.so/034a11e409394585b2b66cc9c3b84363" 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(c|\boldsymbol{x})=\dfrac{P(\boldsymbol{x},c)}{P(\boldsymbol{x})}"><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>c</mi><mi mathvariant="normal">∣</mi><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo separator="true">,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">P(c|\boldsymbol{x})=\dfrac{P(\boldsymbol{x},c)}{P(\boldsymbol{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" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathdefault">c</span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</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:2.363em;vertical-align:-0.936em;"></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" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</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" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></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 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></span></span></span></span></span></p></div></li><li id="https://www.notion.so/90b676a4e10341309b2e3133218ff5f0" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">贝叶斯分类器</span></span><div id="https://www.notion.so/f9a096eb3a7849c4b68f9dcb277e1e2d" 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%2F7c27fa4b-13da-4382-b948-145f88f9d451%2FUntitled.png?width=414.96875&table=block&id=f9a096eb-3a78-49c4-b68f-9dcb277e1e2d"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F7c27fa4b-13da-4382-b948-145f88f9d451%2FUntitled.png?width=414.96875&table=block&id=f9a096eb-3a78-49c4-b68f-9dcb277e1e2d" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div></li></ul></li></ul><div id="https://www.notion.so/84409bc21a7c48ddb7c2c02c33cc95a7" 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%2Fb15bd2ba-b843-42bb-8401-905f2cfdcf14%2FUntitled.png?width=432&table=block&id=84409bc2-1a7c-48dd-b7c2-c02c33cc95a7"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fb15bd2ba-b843-42bb-8401-905f2cfdcf14%2FUntitled.png?width=432&table=block&id=84409bc2-1a7c-48dd-b7c2-c02c33cc95a7" style="width:432px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/18e145cc3d1c434ca63664bf2a907888" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">假设有4个samples：</span></span></p></div><div id="https://www.notion.so/90ec380d63754dd590ad833be2caa48a" 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%2Ff065c292-64d7-4709-8865-aeaea3a89ca1%2FUntitled.png?width=528&table=block&id=90ec380d-6375-4dd5-90ad-833be2caa48a"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Ff065c292-64d7-4709-8865-aeaea3a89ca1%2FUntitled.png?width=528&table=block&id=90ec380d-6375-4dd5-90ad-833be2caa48a" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/2002b56123e44b3883dbc7431a49b259" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">生成模型：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\sum P(x,y)=1"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∑</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\sum P(x,y)=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.00001em;vertical-align:-0.25001em;"></span><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</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">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="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">1</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/7e820b9fd998406c9be402f8ed2ad213" 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%2F520d3c8f-018e-461a-8001-cb39478d67d5%2FUntitled.png?width=480&table=block&id=7e820b9f-d998-406c-9be4-02f8ed2ad213"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F520d3c8f-018e-461a-8001-cb39478d67d5%2FUntitled.png?width=480&table=block&id=7e820b9f-d998-406c-9be4-02f8ed2ad213" style="width:480px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/4551ff5addde4e98ad8b3210284a4bac" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">判别模型：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\sum_yP(y|x)=1"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mo>∑</mo><mi>y</mi></msub><mi>P</mi><mo stretchy="false">(</mo><mi>y</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\sum_yP(y|x)=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.185818em;vertical-align:-0.43581800000000004em;"></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:0.0016819999999999613em;"><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 mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.43581800000000004em;"><span></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 mathdefault" style="margin-right:0.03588em;">y</span><span class="mord">∣</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:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/0514ec13c2bd444b9a249e35e20194a2" 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%2Fa66c6058-45a3-4a59-8c16-4f38e9b3e58b%2FUntitled.png?width=480&table=block&id=0514ec13-c2bd-444b-9a24-9e35e20194a2"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fa66c6058-45a3-4a59-8c16-4f38e9b3e58b%2FUntitled.png?width=480&table=block&id=0514ec13-c2bd-444b-9a24-9e35e20194a2" style="width:480px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><h2 id="https://www.notion.so/b3e5b9a1d1274e47a64a497c675437e8" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--2"><a class="Anchor" href="#https://www.notion.so/b3e5b9a1d1274e47a64a497c675437e8"><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">朴素贝叶斯分类器</span></span></h2><h3 id="https://www.notion.so/9f7b4d50cc1e41d69f627a507175fe3a" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/9f7b4d50cc1e41d69f627a507175fe3a"><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">条件独立假设</span></span></h3><div id="https://www.notion.so/d5aa9ee62d2c4b18a442cab78f2fc5a8" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">给定类标号</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="y"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</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:0.03588em;">y</span></span></span></span></span></span><span class="SemanticString">，朴素贝叶斯分类器在估计类条件概率时</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">假设属性之间条件独立</mark></span><span class="SemanticString">。</span></span></p></div><div id="https://www.notion.so/96336531e00c412490cbb2ef304716fe" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">条件独立假设可以形式化的表达如下：</span></span></p></div><p id="https://www.notion.so/8e9ea28dede0451cae6c837eaa692632" class="Equation" data-latex="P(X\,|\,Y=y)=\prod_{i=1}^nP(x_i\,|\,Y=y)"><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>X</mi><mtext> </mtext><mi mathvariant="normal">∣</mi><mtext> </mtext><mi>Y</mi><mo>=</mo><mi>y</mi><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><mtext> </mtext><mi mathvariant="normal">∣</mi><mtext> </mtext><mi>Y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(X\,|\,Y=y)=\prod_{i=1}^nP(x_i\,|\,Y=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 mathdefault" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">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 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: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">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:0.16666666666666666em;"></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">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 mathdefault" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span></span></p><div id="https://www.notion.so/caeafe24a5c24deeaee9ce55f3f612a7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">其中每个训练样本可用一个属性向量</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="X=(x_1,x_2,x_3,\dots,x_n)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><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 separator="true">,</mo><msub><mi>x</mi><mn>3</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X=(x_1,x_2,x_3,\dots,x_n)</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:0.07847em;">X</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">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="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></span></span></span></span><span class="SemanticString">表示，各个属性之间条件独立。</span></span></p></div><div id="https://www.notion.so/caf96c3d9af24cb28324f7f4259dbab5" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">有了条件独立假设，就不必计算</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="X"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</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:0.07847em;">X</span></span></span></span></span></span><span class="SemanticString">和</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="Y"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</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;">Y</span></span></span></span></span></span><span class="SemanticString">的每一种组合的类条件概率，只需</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">对给定的</mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="Y"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</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;">Y</span></span></span></span></span></mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">，计算每个</mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue"><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></mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">的条件概率</mark></span><span class="SemanticString">。不需要很大的训练集就能获得较好的概率估计，更实用。</span></span></p></div><div id="https://www.notion.so/a7251c11da124b74bb5ae317dc381774" 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(x_i\,|\,Y=y)"><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><mi>i</mi></msub><mtext> </mtext><mi mathvariant="normal">∣</mi><mtext> </mtext><mi>Y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(x_i\,|\,Y=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 mathdefault" style="margin-right:0.13889em;">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:0.31166399999999994em;"><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:0.16666666666666666em;"></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">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 mathdefault" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">一般根据类别</mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="y"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</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:0.03588em;">y</span></span></span></span></span></mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">下包含属性</mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue"><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></mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">的实例的比例来估计</mark></span><span class="SemanticString">。</span></span></p></div><h3 id="https://www.notion.so/2e98563221604449a04fa3ebf3d1dfa4" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/2e98563221604449a04fa3ebf3d1dfa4"><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">朴素贝叶斯分类器表达式</span></span></h3><div id="https://www.notion.so/6cd9070a2cf14224bda5a9b325292d15" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">由于对所有类别来说</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="P(\boldsymbol{x})"><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="bold-italic">x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(\boldsymbol{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" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">相同，因此有朴素贝叶斯分类器的表达式</span></span></p></div><p id="https://www.notion.so/42ee4c76e29d4f4baa0f4c02004a9b35" class="Equation" data-latex="h_{nb}(\boldsymbol{x})=\underset{c\in\mathcal{Y}}{\argmax}\;P(c)\prod_{i=1}^dP(x_i\,|\,c)"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>h</mi><mrow><mi>n</mi><mi>b</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mo>=</mo><mi><munder><mo><mi mathvariant="normal">arg max</mi><mo></mo></mo><mrow><mi>c</mi><mo>∈</mo><mi mathvariant="script">Y</mi></mrow></munder></mi><mtext> </mtext><mi>P</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>d</mi></munderover><mi>P</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mtext> </mtext><mi mathvariant="normal">∣</mi><mtext> </mtext><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h_{nb}(\boldsymbol{x})=\underset{c\in\mathcal{Y}}{\argmax}\;P(c)\prod_{i=1}^dP(x_i\,|\,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"><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 mtight"><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">b</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"><span class="mord boldsymbol">x</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:3.1137820000000005em;vertical-align:-1.277669em;"></span><span class="mord"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.43055999999999983em;"><span style="top:-2.161229em;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">c</span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mathcal mtight" style="margin-right:0.08222em;">Y</span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><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></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.006825em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathdefault">c</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:1.8361130000000003em;"><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">d</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">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:0.16666666666666666em;"></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">c</span><span class="mclose">)</span></span></span></span></span></p><div id="https://www.notion.so/e2e4145a715d4ffa80eea2e5e6b1c56c" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">显然，朴素贝叶斯分类器的训练过程就是</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">基于训练集</mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="D"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</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:0.02778em;">D</span></span></span></span></span></mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">来估计类先验概率</mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="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>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">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" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathdefault">c</span><span class="mclose">)</span></span></span></span></span></mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">，并为每个属性估计条件概率</mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="P(x_i\,|\,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><msub><mi>x</mi><mi>i</mi></msub><mtext> </mtext><mi mathvariant="normal">∣</mi><mtext> </mtext><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(x_i\,|\,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" style="margin-right:0.13889em;">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:0.31166399999999994em;"><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:0.16666666666666666em;"></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">c</span><span class="mclose">)</span></span></span></span></span></mark></span><span class="SemanticString">.</span></span></p></div><div id="https://www.notion.so/c44e62fa42f54200aa4f1ce23e0a8d3a" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">令</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="D_c"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mi>c</mi></msub></mrow><annotation encoding="application/x-tex">D_c</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:0.02778em;">D</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:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">c</span></span></span></span><span 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">表示训练集</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="D"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</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:0.02778em;">D</span></span></span></span></span></span><span class="SemanticString">中第</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">类样本组成的集合，若有充足的独立同分布样本，则可容易地估计出类先验概率：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="P(c)=\dfrac{|D_c|}{|D|}"><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>c</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi mathvariant="normal">∣</mi><msub><mi>D</mi><mi>c</mi></msub><mi mathvariant="normal">∣</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi>D</mi><mi mathvariant="normal">∣</mi></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">P(c)=\dfrac{|D_c|}{|D|}</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">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:2.363em;vertical-align:-0.936em;"></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><span class="mord mathdefault" style="margin-right:0.02778em;">D</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">∣</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:0.151392em;"><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">c</span></span></span></span><span class="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 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></span></span></span></span><span class="SemanticString">.</span></span></p></div><div id="https://www.notion.so/f1550183f85f43af82e82e8463213787" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">估计每个属性的条件概率</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="P(x_i\,|\,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><msub><mi>x</mi><mi>i</mi></msub><mtext> </mtext><mi mathvariant="normal">∣</mi><mtext> </mtext><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(x_i\,|\,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" style="margin-right:0.13889em;">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:0.31166399999999994em;"><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:0.16666666666666666em;"></span><span class="mord">∣</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">：</span></span></p></div><div id="https://www.notion.so/fd3278c430a6429a90477451f6e6099f" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">离散：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="P(x_i\,|\,c)=\dfrac{|D_{c,x_i}|}{|D_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><msub><mi>x</mi><mi>i</mi></msub><mtext> </mtext><mi mathvariant="normal">∣</mi><mtext> </mtext><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi mathvariant="normal">∣</mi><msub><mi>D</mi><mrow><mi>c</mi><mo separator="true">,</mo><msub><mi>x</mi><mi>i</mi></msub></mrow></msub><mi mathvariant="normal">∣</mi></mrow><mrow><mi mathvariant="normal">∣</mi><msub><mi>D</mi><mi>c</mi></msub><mi mathvariant="normal">∣</mi></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">P(x_i\,|\,c)=\dfrac{|D_{c,x_i}|}{|D_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" style="margin-right:0.13889em;">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:0.31166399999999994em;"><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:0.16666666666666666em;"></span><span class="mord">∣</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:2.363em;vertical-align:-0.936em;"></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><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:0.151392em;"><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">c</span></span></span></span><span class="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">∣</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:0.15139200000000003em;"><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">c</span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathdefault mtight">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: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">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></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></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></span></span></span></span></span></p></div><div id="https://www.notion.so/0828ea2684144af79edcf0fcf600419b" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">连续：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="P(x_i\,|\,c)=\dfrac{1}{\sqrt{2\pi}\sigma_{c,i}}\exp\Big(-\dfrac{(x_i-\mu_{c,i})^2}{2\sigma_{c,i}^2}\Big)"><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><mi>i</mi></msub><mtext> </mtext><mi mathvariant="normal">∣</mi><mtext> </mtext><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mrow><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt><msub><mi>σ</mi><mrow><mi>c</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub></mrow></mfrac></mstyle><mi>exp</mi><mo></mo><mo fence="false">(</mo><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><msub><mi>μ</mi><mrow><mi>c</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mrow><mn>2</mn><msubsup><mi>σ</mi><mrow><mi>c</mi><mo separator="true">,</mo><mi>i</mi></mrow><mn>2</mn></msubsup></mrow></mfrac></mstyle><mo fence="false">)</mo></mrow><annotation encoding="application/x-tex">P(x_i\,|\,c)=\dfrac{1}{\sqrt{2\pi}\sigma_{c,i}}\exp\Big(-\dfrac{(x_i-\mu_{c,i})^2}{2\sigma_{c,i}^2}\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 mathdefault" style="margin-right:0.13889em;">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:0.31166399999999994em;"><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:0.16666666666666666em;"></span><span class="mord">∣</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:2.404768em;vertical-align:-1.083328em;"></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:-2.2027799999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.90722em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span></span></span><span style="top:-2.86722em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
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c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
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M834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.13278em;"><span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">σ</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 mtight"><span class="mord mathdefault mtight">c</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></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:1.083328em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">exp</span><span class="mspace" style="margin-right:0.16666666666666666em;"></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><span class="base"><span class="strut" style="height:2.5900799999999995em;vertical-align:-1.0989719999999998em;"></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:1.4911079999999999em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.795908em;"><span style="top:-2.4231360000000004em;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">c</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.0448000000000004em;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:0.4129719999999999em;"><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="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="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">μ</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">c</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="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></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.0989719999999998em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="delimsizing size2">)</span></span></span></span></span></span></span></span></p></div><h3 id="https://www.notion.so/6519844af81848b0af22816ad8bf8cdf" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/6519844af81848b0af22816ad8bf8cdf"><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">拉普拉斯修正</span></span></h3><div id="https://www.notion.so/9e4bdc90ba694d188f602ae2f08ce735" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">若某个属性值在训练集中没有与某个类同时出现过，则直接计算连乘式的概率值为0，导致分类结果显然不合理。为了避免其他属性携带的信息被训练集中未出现的属性值“抹去”，在估计概率值时通常要进行</span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">“拉普拉斯修正”</strong></span><span class="SemanticString">。</span></span></p></div><div id="https://www.notion.so/7418d27749cc4eb59551352010078b5f" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">令</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:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">N</span></span></span></span></span></span><span class="SemanticString">表示数据集</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="D"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</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:0.02778em;">D</span></span></span></span></span></span><span class="SemanticString">中可能的类别数，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="N_i"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>N</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">N_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:0.10903em;">N</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:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">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">表示第</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">个属性可能的取值数，则</span></span></p></div><p id="https://www.notion.so/de4eaf9561d54aa690cbddc248db08b6" class="Equation" data-latex="\begin{aligned}
\hat{P}(c) &= \frac{|D_c|+1}{|D|+N} \\
\hat{P}(x_i\,|\,c) &= \frac{|D_{c,x_i}|+1}{|D_c|+N_i}
\end{aligned}"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mover accent="true"><mi>P</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mrow><mi mathvariant="normal">∣</mi><msub><mi>D</mi><mi>c</mi></msub><mi mathvariant="normal">∣</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi mathvariant="normal">∣</mi><mi>D</mi><mi mathvariant="normal">∣</mi><mo>+</mo><mi>N</mi></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mover accent="true"><mi>P</mi><mo>^</mo></mover><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mtext> </mtext><mi mathvariant="normal">∣</mi><mtext> </mtext><mi>c</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mrow><mi mathvariant="normal">∣</mi><msub><mi>D</mi><mrow><mi>c</mi><mo separator="true">,</mo><msub><mi>x</mi><mi>i</mi></msub></mrow></msub><mi mathvariant="normal">∣</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi mathvariant="normal">∣</mi><msub><mi>D</mi><mi>c</mi></msub><mi mathvariant="normal">∣</mi><mo>+</mo><msub><mi>N</mi><mi>i</mi></msub></mrow></mfrac></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
\hat{P}(c) &= \frac{|D_c|+1}{|D|+N} \\
\hat{P}(x_i\,|\,c) &= \frac{|D_{c,x_i}|+1}{|D_c|+N_i}
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.3260000000000005em;vertical-align:-2.4130000000000003em;"></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:2.9130000000000003em;"><span style="top:-4.913em;"><span class="pstrut" style="height:3.427em;"></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:0.13889em;">P</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="mopen">(</span><span class="mord mathdefault">c</span><span class="mclose">)</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.427em;"></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:0.13889em;">P</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="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="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">c</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4130000000000003em;"><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:2.9130000000000003em;"><span style="top:-4.913em;"><span class="pstrut" style="height:3.427em;"></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="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><span class="mord mathdefault" style="margin-right:0.02778em;">D</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 class="mord mathdefault" style="margin-right:0.10903em;">N</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">∣</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:0.151392em;"><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">c</span></span></span></span><span class="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="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></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></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.427em;"></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="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><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:0.151392em;"><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">c</span></span></span></span><span class="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="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:0.10903em;">N</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:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">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">∣</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:0.15139200000000003em;"><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">c</span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathdefault mtight">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: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">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></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><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></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></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4130000000000003em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><div id="https://www.notion.so/c3d84c3e41b6450cbd3f3dfc3c3aa15d" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">（拉普拉斯修正，实际上是假设了属性与类别均匀分布）</span></span></p></div><div id="https://www.notion.so/e90eba55b6f64204a8ed80993c135d71" class="Divider"></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/eaf7d5607e2c4ca6823909fc63893279" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">若任务对预测速度要求较高：</span></span><div id="https://www.notion.so/362bf3698e3c4c18931490a487baaca4" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">计算所有概率估值存储，使用时“查表”。</span></span></p></div></li><li id="https://www.notion.so/a2f23cbb99324ae6a4e89259a190ffef" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">若任务数据更替频繁：</span></span><div id="https://www.notion.so/71fb74c03c0142c48c895f6ffa54c1f5" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">使用“懒惰学习”，即先不进行任务训练，收到预测请求时再估值。</span></span></p></div></li><li id="https://www.notion.so/0fef02313ecd4adfb75f4ad9d071b0dc" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">若任务数据不断增加：</span></span><div id="https://www.notion.so/d7b914f7236c4990b2c8c8d39f16b96e" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">使用“增量学习”，基于现有估值，对新样本涉及的概率估值进行修正。</span></span></p></div></li></ul><h2 id="https://www.notion.so/b10cfc69c1204468b4c6fca273beb827" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--2"><a class="Anchor" href="#https://www.notion.so/b10cfc69c1204468b4c6fca273beb827"><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">半朴素贝叶斯分类器</span></span></h2><div id="https://www.notion.so/0fb199b797e24f968a639968f95d4996" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">对属性条件独立假设进行一定程度的放松，由此产生了一类称为</span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">“半朴素贝叶斯分类器”</strong></span><span class="SemanticString">。</span></span></p></div><div id="https://www.notion.so/2e5a1c2232a7456cb902384a10efc572" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">半朴素贝叶斯分类器最常用的一种策略：“独依赖估计” (One-Dependent Estimator, ODE) ，假设每个属性在类别之外最多仅依赖一个其他属性，即</span></span></p></div><p id="https://www.notion.so/173fa8e5937145148911186b0940a80c" class="Equation" data-latex="P(c\,|\,\boldsymbol{x})\propto P(c)\prod_{i=1}^dP(x_i\,|\,c,{pa}_i)"><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>c</mi><mtext> </mtext><mi mathvariant="normal">∣</mi><mtext> </mtext><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mo>∝</mo><mi>P</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>d</mi></munderover><mi>P</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mtext> </mtext><mi mathvariant="normal">∣</mi><mtext> </mtext><mi>c</mi><mo separator="true">,</mo><msub><mrow><mi>p</mi><mi>a</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(c\,|\,\boldsymbol{x})\propto P(c)\prod_{i=1}^dP(x_i\,|\,c,{pa}_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" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</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:3.1137820000000005em;vertical-align:-1.277669em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathdefault">c</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:1.8361130000000003em;"><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">d</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">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:0.16666666666666666em;"></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">p</span><span class="mord mathdefault">a</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.21752399999999997em;"><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">i</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="mclose">)</span></span></span></span></span></p><div id="https://www.notion.so/00f20bb38f4348a8af9c54eebcd342c4" 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="{pa}_i"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>p</mi><mi>a</mi></mrow><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">{pa}_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6747em;vertical-align:-0.24414em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">p</span><span class="mord mathdefault">a</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.21752399999999997em;"><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">i</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></span></span></span></span><span class="SemanticString">为属性</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">所依赖的属性，称为</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">的父属性。</span></span></p></div><div id="https://www.notion.so/f1e86eb0114f4e6580ef9acf789d6cbe" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">问题关键转化为</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">如何确定每个属性的父属性</mark></span><span class="SemanticString">。常见方法有：</span></span></p></div><div id="https://www.notion.so/8141c8f283af4636ace8a24871c1cedc" 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%2F0b2502e5-3d8b-4f5b-894c-5323a41c8c05%2FUntitled.png?width=1094&table=block&id=8141c8f2-83af-4636-ace8-a24871c1cedc"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F0b2502e5-3d8b-4f5b-894c-5323a41c8c05%2FUntitled.png?width=1094&table=block&id=8141c8f2-83af-4636-ace8-a24871c1cedc" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><h2 id="https://www.notion.so/f69ca3142d0747d6ab2c321a80377af2" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--2"><a class="Anchor" href="#https://www.notion.so/f69ca3142d0747d6ab2c321a80377af2"><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</span></span></h2><div id="https://www.notion.so/45edbc091bd7452c813e45e73740cd72" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">把某个研究系统中涉及的</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">随机变量</mark></span><span class="SemanticString">，根据是否条件独立绘制在一个</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">有向图</mark></span><span class="SemanticString">中，就形成了贝叶斯网络。</span></span></p></div><div id="https://www.notion.so/e1d3f147125f43cb990a6960b69f6446" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">贝叶斯网络 (Bayesian Network) ，是</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">有向无环图模型</mark></span><span class="SemanticString">，也是一种概率图模型，借由有向无环图DAG得知一组随机变量</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\{X_1,X_2,\dots,X_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><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{X_1,X_2,\dots,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="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="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:0.30110799999999993em;"><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">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" 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="mclose">}</span></span></span></span></span></span><span class="SemanticString">及其</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">组</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">条件概率分布</mark></span><span class="SemanticString">的性质。</span></span></p></div><div id="https://www.notion.so/7a42b0c19ea94fd3bf49d501dff71053" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">一般而言，贝叶斯网络的有向无环图中的</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">结点表示随机变量</mark></span><span class="SemanticString">，它们是可观察到的变量，或隐变量、未知参数等。连接两个结点的箭头代表此</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">两个随机变量是具有因果关系</mark></span><span class="SemanticString">（或非条件独立）。若两个结点间以一个单箭头连接在一起，表示其中一个结点是“因 (parents)”，另一个是“果 (children)”，两结点就会产生一个条件概率值。</span></span></p></div><div id="https://www.notion.so/bde59300695b4ec7a42c5a6bcf0c960c" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">每个结点在给定其直接前驱时，条件独立于其非后继。</span></span></p></div><blockquote id="https://www.notion.so/8961a64a1c31462f84d16c71ed38b289" class="ColorfulBlock ColorfulBlock--ColorDefault Quote"><span class="SemanticStringArray"><span class="SemanticString">概率图模型：用图来表示变量概率依赖关系的理论
- 贝叶斯网络 (Bayesian Network) ：有向图结构表示
- 马尔可夫网络 (Markov Network) ：无向图结构表示
概率图模型包括朴素贝叶斯模型、最大熵模型、隐马尔可夫模型、条件随机场、主题模型等。</span></span></blockquote><ul class="BulletedListWrapper"><li id="https://www.notion.so/d090c2084acc4b8fa6ec475992a6d2e4" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">一个简单的贝叶斯网络</span></span></li></ul><div id="https://www.notion.so/58230b1d51554d7fb184eb125b0b999b" class="ColumnList"><div id="https://www.notion.so/11899b54183b4675a2ddf29a8b82749c" class="Column" style="width:calc((100% - var(--column-spacing) * 1) * 0.75)"><p id="https://www.notion.so/69c1911704b440768a3003e68f6d81f8" class="Equation" data-latex="p(a,b,c)=p(c|a,b)p(b|a)p(a)"><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>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><mi mathvariant="normal">∣</mi><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>b</mi><mi mathvariant="normal">∣</mi><mi>a</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(a,b,c)=p(c|a,b)p(b|a)p(a)</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="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 class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">b</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">a</span><span class="mclose">)</span></span></span></span></span></p></div><div id="https://www.notion.so/01a4575bbd8a4513a6eae2beefb1972e" class="Column" style="width:calc((100% - var(--column-spacing) * 1) * 0.24999999999999994)"><div id="https://www.notion.so/4edc37e0180c41a9aa2ccad00aea2a46" 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%2F71fee165-7318-4f73-8a7b-15035f3fcfb8%2FUntitled.png?width=132&table=block&id=4edc37e0-180c-41a9-aa2c-cad00aea2a46"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F71fee165-7318-4f73-8a7b-15035f3fcfb8%2FUntitled.png?width=132&table=block&id=4edc37e0-180c-41a9-aa2c-cad00aea2a46" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div></div></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/50a9ccaddd444a98a839790798cb171a" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">全连接贝叶斯网络：每一对结点之间都有边连接</span></span><p id="https://www.notion.so/69de5f1951784b60aa9e4f59975c1aa8" class="Equation" data-latex="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)"><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=132.796875&table=block&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=132.796875&table=block&id=512a94c9-8ba9-4be5-a47d-b4569286be00" style="width:132.796875px"/></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&table=block&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&table=block&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&table=block&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&table=block&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&table=block&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&table=block&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&table=block&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&table=block&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 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