Ascent‐based Monte Carlo expectation– maximization

Abstract

Summary. The expectation–maximization (EM) algorithm is a popular tool for maximizing likelihood functions in the presence of missing data. Unfortunately, EM often requires the evaluation of analytically intractable and high dimensional integrals. The Monte Carlo EM (MCEM) algorithm is the natural extension of EM that employs Monte Carlo methods to estimate the relevant integrals. Typically, a very large Monte Carlo sample size is required to estimate these integrals within an acceptable tolerance when the algorithm is near convergence. Even if this sample size were known at the onset of implementation of MCEM, its use throughout all iterations is wasteful, especially when accurate starting values are not available. We propose a data‐driven strategy for controlling Monte Carlo resources in MCEM. The algorithm proposed improves on similar existing methods by recovering EM's ascent (i.e. likelihood increasing) property with high probability, being more robust to the effect of user‐defined inputs and handling classical Monte Carlo and Markov chain Monte Carlo methods within a common framework. Because of the first of these properties we refer to the algorithm as ‘ascent‐based MCEM’. We apply ascent‐based MCEM to a variety of examples, including one where it is used to accelerate the convergence of deterministic EM dramatically.