Transcription of The EM Algorithm
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Example: stock market
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.
until θ(t) converges to a local maxima. ... but can become excruciatingly slow as you approach a local optima. Generally, EM works best when the fraction of missing information is small3 and the dimensionality of the data is not too ... Maximizing Q(θ|θ(t)) w.r.t. θ yields the update equations
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