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 <!DOCTYPE html><html lang="en"><head><meta http-equiv="content-type" content="text/html; charset=utf-8"><meta content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=0" name="viewport"><meta content="yes" name="apple-mobile-web-app-capable"><meta content="black-translucent" name="apple-mobile-web-app-status-bar-style"><meta content="telephone=no" name="format-detection"><meta name="description"><title>EM Algorithm | Welcome To Oa</title><link rel="stylesheet" type="text/css" href="/css/style.css?v=0.0.0"><link rel="stylesheet" type="text/css" href="/css/donate.css"><link rel="stylesheet" type="text/css" href="//cdn.bootcss.com/normalize/6.0.0/normalize.min.css"><link rel="stylesheet" type="text/css" href="//cdn.bootcss.com/pure/0.6.2/pure-min.css"><link rel="stylesheet" type="text/css" href="//cdn.bootcss.com/pure/0.6.2/grids-responsive-min.css"><link rel="stylesheet" href="//cdn.bootcss.com/font-awesome/4.7.0/css/font-awesome.min.css"><script type="text/javascript" src="//cdn.bootcss.com/jquery/3.2.1/jquery.min.js"></script><link rel="Shortcut Icon" type="image/x-icon" href="/favicon.ico"><link rel="apple-touch-icon" href="/apple-touch-icon.png"><link rel="apple-touch-icon-precomposed" href="/apple-touch-icon.png"></head><body><div class="body_container"><div id="header"><div class="site-name"><h1 class="hidden">EM Algorithm</h1><a id="logo" href="/.">Welcome To Oa</a><p class="description"></p></div><div id="nav-menu"><a href="/." class="current"><i class="fa fa-home"> Home</i></a><a href="/archives/"><i class="fa fa-archive"> Archive</i></a><a href="/about/"><i class="fa fa-user"> About</i></a><a href="/guestbook/"><i class="fa fa-comments"> Guestbook</i></a><a href="/atom.xml"><i class="fa fa-rss"> RSS</i></a></div></div><div id="layout" class="pure-g"><div class="pure-u-1 pure-u-md-3-4"><div class="content_container"><div class="post"><h1 class="post-title">EM Algorithm</h1><div class="post-meta">Jan 28, 2019<script src="https://dn-lbstatics.qbox.me/busuanzi/2.3/busuanzi.pure.mini.js" async></script><span id="busuanzi_container_page_pv"> | <span id="busuanzi_value_page_pv"></span><span> Hits</span></span></div><div class="clear"><div id="toc" class="toc-article"><div class="toc-title">Contents</div><ol class="toc"><li class="toc-item toc-level-1"><a class="toc-link" href="#Abstract"><span class="toc-number">1.</span> <span class="toc-text">Abstract</span></a></li><li class="toc-item toc-level-1"><a class="toc-link" href="#Introduction"><span class="toc-number">2.</span> <span class="toc-text">Introduction</span></a><ol class="toc-child"><li class="toc-item toc-level-2"><a class="toc-link" href="#Background"><span class="toc-number">2.1.</span> <span class="toc-text">Background</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#Problem-Settings"><span class="toc-number">2.2.</span> <span class="toc-text">Problem Settings</span></a></li></ol></li><li class="toc-item toc-level-1"><a class="toc-link" href="#Algorithm"><span class="toc-number">3.</span> <span class="toc-text">Algorithm</span></a><ol class="toc-child"><li class="toc-item toc-level-2"><a class="toc-link" href="#Details"><span class="toc-number">3.1.</span> <span class="toc-text">Details</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#Coin-filpping-experiment"><span class="toc-number">3.2.</span> <span class="toc-text">Coin-filpping experiment</span></a></li></ol></li><li class="toc-item toc-level-1"><a class="toc-link" href="#References"><span class="toc-number">4.</span> <span class="toc-text">References</span></a></li></ol></div></div><div class="post-content"><h1 id="Abstract"><a href="#Abstract" class="headerlink" title="Abstract"></a>Abstract</h1><p>EM is an algorithm that enables parameter estimation in probabilistic models with incomplete data.</p>
 <h1 id="Introduction"><a href="#Introduction" class="headerlink" title="Introduction"></a>Introduction</h1><h2 id="Background"><a href="#Background" class="headerlink" title="Background"></a>Background</h2><ul>
-<li>maximum-likelihood estimation(MLE): $argmax<em>\theta L(\theta)=argmax</em>\theta \prod<em>{i=1}^n p(x_i;\theta</em>)$</li>
+<li>maximum-likelihood estimation(MLE): $argmax<em>\theta L(\theta)=argmax</em>\theta \prod_{i=1}^n p(x_i;\theta_i)$</li>
 <li>Jesen-inequality: for a concave function f(x), $f(E(x))&gt;=E(f(x))$;equal when x is a constant</li>
 <li>GMM:$ p ( \mathrm { x  }  ) = \sum <em> { i = 1  } ^ { k  } w </em> { i  } N <em> { i  } \left( \mathrm { x  } ; \mu </em> { i  } , \Sigma <em> { i  } \right )$ s.t. $\sum </em> { i = 1  } ^ { k  } w _ { i  } = 1$</li>
 </ul>
@@ -21,6 +21,7 @@ <h2 id="Coin-filpping-experiment"><a href="#Coin-filpping-experiment" class="hea
 <p>Fristly, we assign $\theta_A$ and $\theta_B$, then calculate the hidden variable distribution based on $\theta_A$, $theta_B$ and the samples.After that we maximize the lower bound of log-likelihood function, get new  $\theta_A$ and $\theta_B$. After 10 runs, $\theta_A=0.80$ and $\theta_B=0.58$</p>
 <h1 id="References"><a href="#References" class="headerlink" title="References"></a>References</h1><ul>
 <li>Do, C. B., &amp; Batzoglou, S. (2008). What is the expectation maximization algorithm?. Nature biotechnology, 26(8), 897.</li>
+<li><a href="https://www.zhihu.com/question/27976634/answer/153567695" target="_blank" rel="noopener">https://www.zhihu.com/question/27976634/answer/153567695</a></li>
 </ul>
 </div><script type="text/javascript" src="/js/share.js?v=0.0.0" async></script><a data-url="https://jarvis.world/EM-Algorithm/" data-id="cjrk7d71400054tihbjw061nz" class="article-share-link">Share</a><div class="tags"><a href="/tags/ml-algo/">ml-algo</a></div><div class="post-nav"><a href="/Summary-Mean-Field-Multi-Agent-Reinforcement-Learning/" class="next">Summary: Mean Field Multi-Agent Reinforcement Learning. ICML(2018)</a></div><div id="container"></div><link rel="stylesheet" href="https://imsun.github.io/gitment/style/default.css"><script src="https://imsun.github.io/gitment/dist/gitment.browser.js"></script><script>var gitment = new Gitment({
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     <loc>https://jarvis.world/EM-Algorithm/</loc>
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         <title>EM Algorithm</title>

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