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EM算法 (查看源代码)

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Arthur Dempster, Nan Laird和Donald Rubin<ref name="Dempster1977">
 
Arthur Dempster, Nan Laird和Donald Rubin<ref name="Dempster1977">
 
{{cite journal
 
{{cite journal
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  |last1=Dempster  |first1= A.P. |author-link1=Arthur P. Dempster
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  |last1=Dempster  |first1= A.P.
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  |last2=Laird |first2=N.M. |author-link2=Nan Laird |last3=Rubin
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  |last2=Laird |first2=N.M.|last3=Rubin
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  |first3=D.B. |author-link3=Donald Rubin
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  |first3=D.B.  
 
  |title=Maximum Likelihood from Incomplete Data via the EM Algorithm
 
  |title=Maximum Likelihood from Incomplete Data via the EM Algorithm
 
  |journal=[[Journal of the Royal Statistical Society, Series B]]
 
  |journal=[[Journal of the Royal Statistical Society, Series B]]
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== 正确性证明 ==
 
== 正确性证明 ==
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期望最大化可以改善<math>Q(\boldsymbol\theta\mid\boldsymbol\theta^{(t)})</math>而不是直接改进<math>\log p(\mathbf{X}\mid\boldsymbol\theta)</math> 。 这里表明对前者的改进意味着对后者的改进。<ref name="Little1987">{{cite book |last1=Little |first1= Roderick J.A. |last2= Rubin |first2= Donald B. |author2-link= Donald Rubin |title= Statistical Analysis with Missing Data |url=https://archive.org/details/statisticalanaly00litt |url-access=limited | series = Wiley Series in Probability and Mathematical Statistics |year= 1987 |publisher= John Wiley & Sons |location= New York |isbn= 978-0-471-80254-9 |pages= [https://archive.org/details/statisticalanaly00litt/page/n145 134]–136}}</ref>
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期望最大化可以改善<math>Q(\boldsymbol\theta\mid\boldsymbol\theta^{(t)})</math>而不是直接改进<math>\log p(\mathbf{X}\mid\boldsymbol\theta)</math> 。 这里表明对前者的改进意味着对后者的改进。<ref name="Little1987">{{cite book |last1=Little |first1= Roderick J.A. |last2= Rubin |first2= Donald B. |title= Statistical Analysis with Missing Data |url=https://archive.org/details/statisticalanaly00litt |url-access=limited | series = Wiley Series in Probability and Mathematical Statistics |year= 1987 |publisher= John Wiley & Sons |location= New York |isbn= 978-0-471-80254-9 |pages= [https://archive.org/details/statisticalanaly00litt/page/n145 134]–136}}</ref>
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== 作为最大化-最大化过程 ==
 
== 作为最大化-最大化过程 ==
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EM算法可以看作是两个交替的最大化步骤，即作为坐标下降法的一个例子。<ref name="neal1999">{{cite book|last1=Neal |first=Radford |last2=Hinton |first2=Geoffrey |author-link2=Geoffrey Hinton |title=A view of the EM algorithm that justifies incremental, sparse, and other variants |journal=Learning in Graphical Models |editor=Michael I. Jordan |editor-link=Michael I. Jordan |pages= 355–368 |publisher= MIT Press |location=Cambridge, MA |year=1999 |isbn=978-0-262-60032-3 |url=http://ftp.cs.toronto.edu/pub/radford/emk.pdf |access-date=2009-03-22}}</ref><ref name="hastie2001">{{cite book|last1=Hastie |first1= Trevor|author-link1=Trevor Hastie|last2=Tibshirani|first2=Robert|author-link2=Robert Tibshirani|last3=Friedman|first3=Jerome |year=2001 |title=The Elements of Statistical Learning |url=https://archive.org/details/elementsstatisti00thas_842 |url-access=limited |isbn=978-0-387-95284-0 |publisher=Springer |location=New York |chapter=8.5 The EM algorithm |pages=[https://archive.org/details/elementsstatisti00thas_842/page/n237 236]–243}}</ref>考虑函数：
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EM算法可以看作是两个交替的最大化步骤，即作为坐标下降法的一个例子。<ref name="neal1999">{{cite book|last1=Neal |first=Radford |last2=Hinton |first2=Geoffrey |title=A view of the EM algorithm that justifies incremental, sparse, and other variants |journal=Learning in Graphical Models |editor=Michael I. Jordan |pages= 355–368 |publisher= MIT Press |location=Cambridge, MA |year=1999 |isbn=978-0-262-60032-3 |url=http://ftp.cs.toronto.edu/pub/radford/emk.pdf |access-date=2009-03-22}}</ref><ref name="hastie2001">{{cite book|last1=Hastie |first1= Trevor|last2=Tibshirani|first2=Robert|last3=Friedman|first3=Jerome |year=2001 |title=The Elements of Statistical Learning |url=https://archive.org/details/elementsstatisti00thas_842 |url-access=limited |isbn=978-0-387-95284-0 |publisher=Springer |location=New York |chapter=8.5 The EM algorithm |pages=[https://archive.org/details/elementsstatisti00thas_842/page/n237 236]–243}}</ref>考虑函数：
    
:<math> F(q,\theta) := \operatorname{E}_q [ \log L (\theta ; x,Z) ] + H(q), </math>
 
:<math> F(q,\theta) := \operatorname{E}_q [ \log L (\theta ; x,Z) ] + H(q), </math>
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|doi = 10.1109/TSP.2008.2007090
 
|doi = 10.1109/TSP.2008.2007090
 
|bibcode = 2009ITSP...57..370E
 
|bibcode = 2009ITSP...57..370E
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|s2cid = 1930004
   
}}</ref><ref>{{Cite journal
 
}}</ref><ref>{{Cite journal
 
|last1 = Einicke
 
|last1 = Einicke
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|date=May 2010
 
|date=May 2010
 
|doi = 10.1109/LSP.2010.2043151
 
|doi = 10.1109/LSP.2010.2043151
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|bibcode = 2010ISPL...17..437E |s2cid = 14114266
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|bibcode = 2010ISPL...17..437E  
 
}}</ref><ref>{{Cite journal
 
}}</ref><ref>{{Cite journal
 
|last1 = Einicke
 
|last1 = Einicke
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|bibcode = 2012ISPL...19..275E
 
|bibcode = 2012ISPL...19..275E
 
|doi = 10.1109/LSP.2012.2190278
 
|doi = 10.1109/LSP.2012.2190278
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|s2cid = 17476971
   
}}</ref><ref>{{Cite journal
 
}}</ref><ref>{{Cite journal
 
|last = Einicke
 
|last = Einicke
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|doi = 10.1109/TAES.2015.140843
 
|doi = 10.1109/TAES.2015.140843
 
|bibcode = 2015ITAES..51.2205E
 
|bibcode = 2015ITAES..51.2205E
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|s2cid = 32667132
   
}}</ref>
 
}}</ref>
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''期望条件最大化 Expectation conditional maximization (ECM)''用一系列条件最大化(CM)步骤代替每个 m 步骤，其中每个参数''θ''<sub>''i''</sub>单独最大化，条件是其他参数保持不变。<ref>{{cite journal|last1=Meng  |first1= Xiao-Li |last2=Rubin |first2=Donald B. |s2cid= 40571416 |author-link2=Donald Rubin |title=Maximum likelihood estimation via the ECM algorithm: A general framework |year=1993 |journal=[[Biometrika]] |volume=80 |issue=2 |pages=267–278 |doi=10.1093/biomet/80.2.267 |mr=1243503}}</ref> 本身也可以扩展为期望条件最大化 Expectation conditional maximization either(ECME)算法。<ref>{{cite journal |doi= 10.1093/biomet/81.4.633|jstor=2337067 |title=The ECME Algorithm: A Simple Extension of EM and ECM with Faster Monotone Convergence |journal=Biometrika |volume=81 |issue=4 |pages=633 |year=1994 |last1=Liu |first1=Chuanhai |last2=Rubin |first2=Donald B }}</ref>
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''期望条件最大化 Expectation conditional maximization (ECM)''用一系列条件最大化(CM)步骤代替每个 m 步骤，其中每个参数''θ''<sub>''i''</sub>单独最大化，条件是其他参数保持不变。<ref>{{cite journal|last1=Meng  |first1= Xiao-Li |last2=Rubin |first2=Donald B. |title=Maximum likelihood estimation via the ECM algorithm: A general framework |year=1993 |journal=[[Biometrika]] |volume=80 |issue=2 |pages=267–278 |doi=10.1093/biomet/80.2.267 |mr=1243503}}</ref> 本身也可以扩展为期望条件最大化 Expectation conditional maximization either(ECME)算法。<ref>{{cite journal |doi= 10.1093/biomet/81.4.633|jstor=2337067 |title=The ECME Algorithm: A Simple Extension of EM and ECM with Faster Monotone Convergence |journal=Biometrika |volume=81 |issue=4 |pages=633 |year=1994 |last1=Liu |first1=Chuanhai |last2=Rubin |first2=Donald B }}</ref>
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==进一步阅读==
 
==进一步阅读==
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* {{cite book |first1=Robert |last1=Hogg |first2=Joseph |last2=McKean |author-link3=Allen Craig |first3=Allen |last3=Craig |title=Introduction to Mathematical Statistics |pages=359–364 |location=Upper Saddle River, NJ |publisher=Pearson Prentice Hall |year=2005 }}
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* {{cite book |first1=Robert |last1=Hogg |first2=Joseph |last2=McKean |first3=Allen |last3=Craig |title=Introduction to Mathematical Statistics |pages=359–364 |location=Upper Saddle River, NJ |publisher=Pearson Prentice Hall |year=2005 }}
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* {{cite journal |citeseerx= 10.1.1.9.9735 |title= The Expectation Maximization Algorithm|first=Frank |last= Dellaert |author-link= Frank Dellaert |year= 2002}} gives an easier explanation of EM algorithm as to lowerbound maximization.
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* {{cite journal |citeseerx= 10.1.1.9.9735 |title= The Expectation Maximization Algorithm|first=Frank |last= Dellaert |year= 2002}} gives an easier explanation of EM algorithm as to lowerbound maximization.
 
* {{cite book
 
* {{cite book
 
  |last1 = Bishop |first1 = Christopher M.
 
  |last1 = Bishop |first1 = Christopher M.
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|author-link = Christopher Bishop
   
  |title = Pattern Recognition and Machine Learning
 
  |title = Pattern Recognition and Machine Learning
 
  |year = 2006
 
  |year = 2006
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