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<html lang="en"><head><meta charset="utf-8"><meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=1"><meta><title>VAE：变分推断、MC估计和重要性采样 - Rick Universe</title><link rel="manifest" href="/manifest.json"><meta name="application-name" content="Rick Universe"><meta name="msapplication-TileImage" content="/img/agumon.svg"><meta name="apple-mobile-web-app-capable" content="yes"><meta name="apple-mobile-web-app-title" content="Rick Universe"><meta name="apple-mobile-web-app-status-bar-style" content="default"><meta name="description" content="本文将从表征学习的角度以及隐变量模型的角度探讨VAE模型 隐变量的意义 VAE从概率密度估计的角度来说是一种mixture model 基本的高斯混合模型写做： \(p(x) &amp;#x3D; \sum_{i&amp;#x3D;1}^{k} w_i \mathcal{N}(x | \mu_i, \Sigma_i)\) 其中的i是指第i个高斯分布，\(u_i\)和\(\Sigma_i\)​指的是第i个高斯分布的"><meta property="og:type" content="blog"><meta property="og:title" content="VAE：变分推断、MC估计和重要性采样"><meta property="og:url" content="https://rickustc.github.io/2024/04/26/VAE%EF%BC%9A%E5%8F%98%E5%88%86%E6%8E%A8%E6%96%AD%E3%80%81MC%E4%BC%B0%E8%AE%A1%E5%92%8C%E9%87%8D%E8%A6%81%E6%80%A7%E9%87%87%E6%A0%B7/"><meta property="og:site_name" content="Rick Universe"><meta property="og:description" content="本文将从表征学习的角度以及隐变量模型的角度探讨VAE模型 隐变量的意义 VAE从概率密度估计的角度来说是一种mixture model 基本的高斯混合模型写做： \(p(x) &amp;#x3D; \sum_{i&amp;#x3D;1}^{k} w_i \mathcal{N}(x | \mu_i, \Sigma_i)\) 其中的i是指第i个高斯分布，\(u_i\)和\(\Sigma_i\)​指的是第i个高斯分布的"><meta property="og:locale" content="en_US"><meta property="og:image" content="d:/Blog/blog/source/image/image-20240426131714727.png"><meta property="og:image" content="d:/Blog/blog/source/image/image-20240426152338348.png"><meta property="article:published_time" content="2024-04-25T16:00:00.000Z"><meta property="article:modified_time" content="2024-04-26T07:40:57.099Z"><meta property="article:author" content="Rick Wang"><meta property="article:tag" content="DGM"><meta property="article:tag" content="LVM"><meta property="article:tag" content="VAE"><meta property="twitter:card" content="summary"><meta property="twitter:image:src" content="d:/Blog/blog/source/image/image-20240426131714727.png"><script type="application/ld+json">{"@context":"https://schema.org","@type":"BlogPosting","mainEntityOfPage":{"@type":"WebPage","@id":"https://rickustc.github.io/2024/04/26/VAE%EF%BC%9A%E5%8F%98%E5%88%86%E6%8E%A8%E6%96%AD%E3%80%81MC%E4%BC%B0%E8%AE%A1%E5%92%8C%E9%87%8D%E8%A6%81%E6%80%A7%E9%87%87%E6%A0%B7/"},"headline":"VAE：变分推断、MC估计和重要性采样","image":["d:/Blog/blog/source/image/image-20240426131714727.png","d:/Blog/blog/source/image/image-20240426152338348.png"],"datePublished":"2024-04-25T16:00:00.000Z","dateModified":"2024-04-26T07:40:57.099Z","author":{"@type":"Person","name":"Rick Wang"},"publisher":{"@type":"Organization","name":"Rick Universe","logo":{"@type":"ImageObject","url":"https://rickustc.github.io/img/agumon.svg"}},"description":"本文将从表征学习的角度以及隐变量模型的角度探讨VAE模型\r \r 隐变量的意义\r VAE从概率密度估计的角度来说是一种mixture model\r 基本的高斯混合模型写做：\r \\(p(x) &#x3D; \\sum_{i&#x3D;1}^{k} w_i \\mathcal{N}(x |\r \\mu_i, \\Sigma_i)\\)\r 其中的i是指第i个高斯分布，\\(u_i\\)和\\(\\Sigma_i\\)​指的是第i个高斯分布的"}</script><link rel="canonical" href="https://rickustc.github.io/2024/04/26/VAE%EF%BC%9A%E5%8F%98%E5%88%86%E6%8E%A8%E6%96%AD%E3%80%81MC%E4%BC%B0%E8%AE%A1%E5%92%8C%E9%87%8D%E8%A6%81%E6%80%A7%E9%87%87%E6%A0%B7/"><link rel="icon" href="/img/agumon.svg"><link rel="stylesheet" href="https://use.fontawesome.com/releases/v6.0.0/css/all.css"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/[email protected]/styles/atom-one-light.css"><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=Ubuntu:wght@400;600&amp;family=Source+Code+Pro"><link rel="stylesheet" href="/css/default.css"><style>body>.footer,body>.navbar,body>.section{opacity:0}</style><!--!--><!--!--><!--!--><script src="//busuanzi.ibruce.info/busuanzi/2.3/busuanzi.pure.mini.js" defer></script><!--!--><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/[email protected]/build/cookieconsent.min.css"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/[email protected]/dist/css/lightgallery.min.css"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/[email protected]/dist/css/justifiedGallery.min.css"><!--!--><!--!--><!--!--><style>.pace{-webkit-pointer-events:none;pointer-events:none;-webkit-user-select:none;-moz-user-select:none;user-select:none}.pace-inactive{display:none}.pace .pace-progress{background:#3273dc;position:fixed;z-index:2000;top:0;right:100%;width:100%;height:2px}</style><script src="https://cdn.jsdelivr.net/npm/[email protected]/pace.min.js"></script><!--!--><!--!--><!-- hexo injector head_end start -->
<html lang="en"><head><meta charset="utf-8"><meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=1"><meta><title>VAE：变分推断、MC估计和重要性采样 - Rick Universe</title><link rel="manifest" href="/manifest.json"><meta name="application-name" content="Rick Universe"><meta name="msapplication-TileImage" content="/img/agumon.svg"><meta name="apple-mobile-web-app-capable" content="yes"><meta name="apple-mobile-web-app-title" content="Rick Universe"><meta name="apple-mobile-web-app-status-bar-style" content="default"><meta name="description" content="本文将从表征学习的角度以及隐变量模型的角度探讨VAE模型 隐变量的意义 VAE从概率密度估计的角度来说是一种mixture model 基本的高斯混合模型写做： \(p(x) &amp;#x3D; \sum_{i&amp;#x3D;1}^{k} w_i \mathcal{N}(x | \mu_i, \Sigma_i)\) 其中的i是指第i个高斯分布，\(u_i\)和\(\Sigma_i\)​指的是第i个高斯分布的"><meta property="og:type" content="blog"><meta property="og:title" content="VAE：变分推断、MC估计和重要性采样"><meta property="og:url" content="https://rickustc.github.io/2024/04/26/VAE%EF%BC%9A%E5%8F%98%E5%88%86%E6%8E%A8%E6%96%AD%E3%80%81MC%E4%BC%B0%E8%AE%A1%E5%92%8C%E9%87%8D%E8%A6%81%E6%80%A7%E9%87%87%E6%A0%B7/"><meta property="og:site_name" content="Rick Universe"><meta property="og:description" content="本文将从表征学习的角度以及隐变量模型的角度探讨VAE模型 隐变量的意义 VAE从概率密度估计的角度来说是一种mixture model 基本的高斯混合模型写做： \(p(x) &amp;#x3D; \sum_{i&amp;#x3D;1}^{k} w_i \mathcal{N}(x | \mu_i, \Sigma_i)\) 其中的i是指第i个高斯分布，\(u_i\)和\(\Sigma_i\)​指的是第i个高斯分布的"><meta property="og:locale" content="en_US"><meta property="og:image" content="d:/Blog/blog/source/image/image-20240426131714727.png"><meta property="og:image" content="d:/Blog/blog/source/image/image-20240426152338348.png"><meta property="article:published_time" content="2024-04-25T16:00:00.000Z"><meta property="article:modified_time" content="2024-04-26T07:48:15.146Z"><meta property="article:author" content="Rick Wang"><meta property="article:tag" content="DGM"><meta property="article:tag" content="LVM"><meta property="article:tag" content="VAE"><meta property="twitter:card" content="summary"><meta property="twitter:image:src" content="d:/Blog/blog/source/image/image-20240426131714727.png"><script type="application/ld+json">{"@context":"https://schema.org","@type":"BlogPosting","mainEntityOfPage":{"@type":"WebPage","@id":"https://rickustc.github.io/2024/04/26/VAE%EF%BC%9A%E5%8F%98%E5%88%86%E6%8E%A8%E6%96%AD%E3%80%81MC%E4%BC%B0%E8%AE%A1%E5%92%8C%E9%87%8D%E8%A6%81%E6%80%A7%E9%87%87%E6%A0%B7/"},"headline":"VAE：变分推断、MC估计和重要性采样","image":["d:/Blog/blog/source/image/image-20240426131714727.png","d:/Blog/blog/source/image/image-20240426152338348.png"],"datePublished":"2024-04-25T16:00:00.000Z","dateModified":"2024-04-26T07:48:15.146Z","author":{"@type":"Person","name":"Rick Wang"},"publisher":{"@type":"Organization","name":"Rick Universe","logo":{"@type":"ImageObject","url":"https://rickustc.github.io/img/agumon.svg"}},"description":"本文将从表征学习的角度以及隐变量模型的角度探讨VAE模型\r \r 隐变量的意义\r VAE从概率密度估计的角度来说是一种mixture model\r 基本的高斯混合模型写做：\r \\(p(x) &#x3D; \\sum_{i&#x3D;1}^{k} w_i \\mathcal{N}(x |\r \\mu_i, \\Sigma_i)\\)\r 其中的i是指第i个高斯分布，\\(u_i\\)和\\(\\Sigma_i\\)​指的是第i个高斯分布的"}</script><link rel="canonical" href="https://rickustc.github.io/2024/04/26/VAE%EF%BC%9A%E5%8F%98%E5%88%86%E6%8E%A8%E6%96%AD%E3%80%81MC%E4%BC%B0%E8%AE%A1%E5%92%8C%E9%87%8D%E8%A6%81%E6%80%A7%E9%87%87%E6%A0%B7/"><link rel="icon" href="/img/agumon.svg"><link rel="stylesheet" href="https://use.fontawesome.com/releases/v6.0.0/css/all.css"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/[email protected]/styles/atom-one-light.css"><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=Ubuntu:wght@400;600&amp;family=Source+Code+Pro"><link rel="stylesheet" href="/css/default.css"><style>body>.footer,body>.navbar,body>.section{opacity:0}</style><!--!--><!--!--><!--!--><script src="//busuanzi.ibruce.info/busuanzi/2.3/busuanzi.pure.mini.js" defer></script><!--!--><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/[email protected]/build/cookieconsent.min.css"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/[email protected]/dist/css/lightgallery.min.css"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/[email protected]/dist/css/justifiedGallery.min.css"><!--!--><!--!--><!--!--><style>.pace{-webkit-pointer-events:none;pointer-events:none;-webkit-user-select:none;-moz-user-select:none;user-select:none}.pace-inactive{display:none}.pace .pace-progress{background:#3273dc;position:fixed;z-index:2000;top:0;right:100%;width:100%;height:2px}</style><script src="https://cdn.jsdelivr.net/npm/[email protected]/pace.min.js"></script><!--!--><!--!--><!-- hexo injector head_end start -->
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</script><!-- hexo injector head_end end --><meta name="generator" content="Hexo 6.3.0"></head><body class="is-3-column"><nav class="navbar navbar-main"><div class="container navbar-container"><div class="navbar-brand justify-content-center"><a class="navbar-item navbar-logo" href="/"><img src="/img/agumon.svg" alt="Rick Universe" height="28"></a></div><div class="navbar-menu"><div class="navbar-start"><a class="navbar-item" href="/rick">RICK UNIVERSE</a><a class="navbar-item" href="/curriculum">Curriculum</a><a class="navbar-item" href="/">BLOG</a><a class="navbar-item" href="/life">LIFE</a><a class="navbar-item" href="/archives">ARCHIVES</a><a class="navbar-item" href="/categories">CATEGORIES</a><a class="navbar-item" href="/tags">TAGS</a><a class="navbar-item" href="/about">ABOUT</a></div><div class="navbar-end"><a class="navbar-item" target="_blank" rel="noopener" title="Download on GitHub" href="https://github.com/ppoffice/hexo-theme-icarus"><i class="fab fa-github"></i></a><a class="navbar-item search" title="Search" href="javascript:;"><i class="fas fa-search"></i></a></div></div></div></nav><section class="section"><div class="container"><div class="columns"><div class="column order-2 column-main is-8-tablet is-8-desktop is-6-widescreen"><div class="card"><article class="card-content article" role="article"><div class="article-meta is-size-7 is-uppercase level is-mobile"><div class="level-left"><span class="level-item">Posted&nbsp;<time dateTime="2024-04-25T16:00:00.000Z" title="2024/4/26上午12:00:00">2024-04-26</time></span><span class="level-item">Updated&nbsp;<time dateTime="2024-04-26T07:48:15.146Z" title="2024/4/26下午3:48:15">2024-04-26</time></span><span class="level-item">9 minutes read (About 1336 words)</span><span class="level-item" id="busuanzi_container_page_pv"><span id="busuanzi_value_page_pv">0</span>&nbsp;visits</span></div></div><h1 class="title is-3 is-size-4-mobile">VAE：变分推断、MC估计和重要性采样</h1><div class="content"><blockquote>
<p>本文将从表征学习的角度以及隐变量模型的角度探讨VAE模型</p>
</blockquote>
<h1 id="隐变量的意义">隐变量的意义</h1>
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<figure>
<img
src="D:\Blog\blog\source\image\image-20240426152256850-1714116180351-5.png"
alt="image-20240426152256850" />
title="高斯混合模型" alt="image-20240426152256850" />
<figcaption aria-hidden="true">image-20240426152256850</figcaption>
</figure>
<figure>
<img src="D:\Blog\blog\source\image\image-20240426152338348.png"
alt="image-20240426152338348" />
title="重要性采样" alt="image-20240426152338348" />
<figcaption aria-hidden="true">image-20240426152338348</figcaption>
</figure>
<p>在重要性采样的语境下，<span
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