Maximum Likelihood (ML), Expectation Maximization (EM)

Maximum Likelihood (ML), Expectation Maximization (EM) Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAA n Maximum Likelihood (ML) n Priors, and Maximum a posteriori (MAP) n Cross-validation n Expectation Maximization (EM) Outline n Let = P(up), 1- = P(down) n How to determine ? n Empirical estimate: 8 up, 2 down Thumbtack n n = P(up), 1- = P(down) n Observe: n Likelihood of the observation sequence depends on : n Maximum Likelihood finds extrema at = 0, = 1, = Inspection of each extremum yields ML = Maximum Likelihood n More generally, consider binary-valued random variable with = P(1), 1- = P(0), assume we observe n1 ones, and n0 zeros n Likelihood : n Derivative: n Hence we have for the extrema: n n1/(n0+n1) is the Maximum n = empirical counts.

Find maximum likelihood estimates of µ 1, µ 2 ! EM basic idea: if x(i) were known " two easy-to-solve separate ML problems ! EM iterates over ! E-step: For i=1,…,m fill in missing data x(i) according to what is most likely given the current model µ ! M-step: run ML for completed data, which gives new model µ

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  Maximum, Maximization, Estimates, Expectations, Likelihood, Maximum likelihood, Expectation maximization, Maximum likelihood estimates

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