The Expectation Maximization Algorithm

This note represents my attempt at explaining the EM algorithm (Hartley, 1958; Dempster et al., 1977; McLachlan and Krishnan, 1997). This is just a slight variation on TomMinka’s tutorial (Minka, 1998), perhaps a little easier (or perhaps not). It includes a graphical example to provide some intuition. 1 Intuitive Explanation of EM EM is an iterative optimizationmethod to estimate some unknown parametersΘ, given measurement dataU. However, we are not given some “hidden” nuisance variables J, which need to be integrated out. In particular, we want to maximize the posterior probability of the parametersΘ given the dataU, marginalizing over J: Θ∗ = argmax Θ ∑ J∈Jn P (Θ,J|U) (1) The intuition behind EM is an old one: alternate between estimating the unknowns Θ and the hidden variables J. This idea has been around for a long time. However, instead of finding the best J ∈ J given an estimateΘ at each iteration, EM computes a distribution over the space J . One of the earliest papers on EM is (Hartley, 1958), but the seminal reference that formalized EM and provided a proof of convergence is the “DLR” paper by Dempster, Laird, and Rubin (Dempster et al., 1977). A recent book devoted entirely to EM and applications is (McLachlan and Krishnan, 1997), whereas (Tanner, 1996) is another popular and very useful reference. One of the most insightful explanations of EM, that provides a deeper understanding of its operation than the intuition of alternating between variables, is in terms of lowerbound maximization (Neal and Hinton, 1998; Minka, 1998). In this derivation, the E-step can be interpreted as constructing a local lower-bound to the posterior distribution, whereas the M-step optimizes the bound, thereby improving the estimate for the unknowns. This is demonstrated below for a simple example. 1 −5 −4 −3 −2 −1 0 1 2 3 4 5 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 1: EM example: Mixture components and data. The data consists of three samples drawn from each mixture component, shown above as circles and triangles. The means of the mixture components are −2 and 2, respectively.