Convergence of a stochastic approximation version of the EM algorithm

TLDR

The EM algorithm is a powerful technique for locating maxima of functions and is widely used for maximum likelihood or maximum a posteriori estimation in incomplete data models, but it fails when the expectation step cannot be computed in closed form. The authors introduce the stochastic approximation EM (SAEM) to address situations where the EM expectation step cannot be performed in closed form. SAEM replaces the EM expectation step with a single iteration of a stochastic approximation procedure. The SAEM algorithm converges under broadly applicable conditions, and its stationary points correspond to local maxima of the target function.

Abstract

The expectation-maximization (EM) algorithm is a powerful computational technique for locating maxima of functions. It is widely used in statistics for maximum likelihood or maximum a posteriori estimation in incomplete data models. In certain situations, however, this method is not applicable because the expectation step cannot be performed in closed form. To deal with these problems, a novel method is introduced, the stochastic approximation EM (SAEM), which replaces the expectation step of the EM algorithm by one iteration of a stochastic approximation procedure. The convergence of the SAEM algorithm is established under conditions that are applicable to many practical situations. Moreover, it is proved that, under mild additional conditions, the attractive stationary points of the SAEM algorithm correspond to the local maxima of the function presented to support our findings.