A gentle tutorial of the em algorithm and its application to parameter estimation for Gaussian mixture and hidden Markov models

Abstract

We describe the maximum-likelihood parameter estimation problem and how the ExpectationMaximization (EM) algorithm can be used for its solution. We first describe the abstract form of the EM algorithm as it is often given in the literature. We then develop the EM parameter estimation procedure for two applications: 1) finding the parameters of a mixture of Gaussian 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 convergence properties. We try to emphasize intuition rather than mathematical rigor. ii 1 Maximum-likelihood Recall the definition of the maximum-likelihood estimation problem. We have a density function p#xj## that is governed by the set of parameters # (e.g., p might be a set of Gaussians and # could be the means and covariances). We also have a data set of size N , supposedl...