Coupling a stochastic approximation version of EM with an MCMC procedure

¹ Université Paris Sud, Bât. 425, 91400 Orsay, France; Estelle.Kuhn@math.u-psud.fr.
² Université René Descartes and Université Paris Sud, Bât. 425, 91400 Orsay, France; Marc.Lavielle@math.u-psud.fr.

Received: 4 May 2003
Revised: 2 November 2003

Abstract

The stochastic approximation version of EM (SAEM) proposed by Delyon et al. (1999) is a powerful alternative to EM when the E-step is intractable. Convergence of SAEM toward a maximum of the observed likelihood is established when the unobserved data are simulated at each iteration under the conditional distribution. We show that this very restrictive assumption can be weakened. Indeed, the results of Benveniste et al. for stochastic approximation with Markovian perturbations are used to establish the convergence of SAEM when it is coupled with a Markov chain Monte-Carlo procedure. This result is very useful for many practical applications. Applications to the convolution model and the change-points model are presented to illustrate the proposed method.