The EM Algorithm
The EM AlgorithmAjit SinghNovember 20, 20051 IntroductionExpectation- maximization (EM) is a technique used in point estimation. Given a set of observablevariablesXand unknown (latent) variablesZwe want to estimate parameters in a (Binomial Mixture Model).You have two coins with unknown probabilities ofheads, denotedpandqrespectively. The first coin is chosen with probability and the secondcoin is chosen with probability 1 . The chosen coin is flipped once and the result is {1,1,0,1,0,0,1,0,0,0,1,1}(Heads = 1, Tails = 0). LetZi {0,1}denote which coin wasused on each example we added latent variablesZifor reasons that will become apparent. The parameterswe want to estimate are = (p,q, ). Two criteria for point estimation are maximum likelihoodand maximum a posteriori: ML= arg max logp(x| ) MAP= arg max logp(x, )= arg max [logp(x| ) + logp( )]Our presentation will focus on the maximum likelihood case (ML-EM); the maximum a posterioricase (MAP-EM) is very NotationXObserved variablesZLatent (unobserved) variables (t)The estimate of the parameters at iterationt.
The EM Algorithm Ajit Singh November 20, 2005 1 Introduction Expectation-Maximization (EM) is a technique used in point estimation. Given a set of observable variables X and unknown (latent) variables Z we want to estimate parameters θ in a model. Example 1.1 (Binomial Mixture Model). You have two coins with unknown probabilities of
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