em-tutorial

NPO>3Q/>=M,%('>)>A('(RS$B$&%BTU8EVXWY$&A>Z[?('*\RX%Y$]W>\U8E^7_a`bVXWY$&A>Z[?('*\RX)>?BT('12-@?>ABAC8ED.-03GFH-I !=!"#-J&KL+.M>I$&%('*),+.-0/12-354*17698:4<;>=A Gentle Tutorial of the EM Algorithmand its Application to ParameterEstimation for Gaussian Mixture andHidden Markov ModelsJeff A. Bilmes (bilmes@cs.berkeley.edu)International Computer Science InstituteBerkeley CA, 94704andComputer Science DivisionDepartment of Electrical Engineering and Computer ScienceU.C. BerkeleyTR-97-021April 1998AbstractWe describe the maximum-likelihood parameter estimation problem and how the Expectation-Maximization (EM) algorithm can be used for its solution. We first describe the abstractform of the EM algorithm as it is often given in the literature. We then develop the EM pa-rameter estimation procedure for two applications: 1) finding the parameters of a mixture ofGaussian densities, and 2) finding the parameters of a hidden Markov model (HMM) (i.e.,the Baum-Welch algorithm) for both discrete and Gaussian mixture observation models.We derive the update equations in fairly explicit detail but we do not prove any conver-gence properties. We try to emphasize intuition rather than mathematical rigor.ii#$45%&.6! 789:;'(& :<9=> 10/)+*-,!.'(! ...