The Stochastic EM Algorithm: Estimation and Asymptotic Results

Abstract

The EM algorithm (Dempster, Laird, and Rubin (1978)) is a much used tool for maximum likelihood estimation in missing or incomplete data problems. However, calculating the conditional expectation required in the E-step of the algorithm may be infeasible, especially when this expectation is a large sum or a high-dimensional integral. Instead an estimate of the expectation can be formed by simulation. This is the common idea in the stochastic EM algorithm (Celeux and Diebolt (1986)) and the Monte Carlo EM algorithm (Wei and Tanner (1990)).In this paper some asymptotic results for the stochastic EM algorithm are given, and estimation based on the stochastic EM algorithm is discussed. In particular, asymptotic equivalence of certain simple estimators is shown, and a simulation experiment is carried out to investigate this equivalence in small and moderate samples. Furthermore, some implementation issues and the possibility of allowing unidenti ed parameters in the algorithm are discussed.

Language Evolution by Iterated Learning With Bayesian Agents

“…The additional variability introduced by sampling in stochastic EM means that it is less likely to get stuck in sub-optimal solutions, although the output of the algorithm is a distribution over hypotheses rather than a single hypothesis. The sequence of hypotheses produced by stochastic EM form a homogeneous Markov chain, and conditions for the ergodicity of this chain have been established (Ip, 1994(Ip, , 2002Diebolt & Ip, 1996;Nielsen, 2000). Empirical and theoretical results indicate that the stationary distribution over hypotheses produced by this Markov chain is approximately centered on the maximum-likelihood solution, with a variance that increases as a function of the rate at which the hypotheses change across iterations (Celeux & Diebolt, 1985, 1988Celeux, Chauveau, & Diebolt, 1995;Diebolt & Ip, 1996;Ip, 1994;Nielsen, 2000).…”

“…The sequence of hypotheses produced by stochastic EM form a homogeneous Markov chain, and conditions for the ergodicity of this chain have been established (Ip, 1994(Ip, , 2002Diebolt & Ip, 1996;Nielsen, 2000). Empirical and theoretical results indicate that the stationary distribution over hypotheses produced by this Markov chain is approximately centered on the maximum-likelihood solution, with a variance that increases as a function of the rate at which the hypotheses change across iterations (Celeux & Diebolt, 1985, 1988Celeux, Chauveau, & Diebolt, 1995;Diebolt & Ip, 1996;Ip, 1994;Nielsen, 2000). A more precise characterization of the consequences of stochastic EM can be given in special cases, such as estimating parameters for the kind of clustering problem introduced above (Diebolt & Celeux, 1993;Nielsen, 2000), but there are no explicit results characterizing the asymptotic behavior of stochastic EM when the set of hypotheses is discrete.…”

Monte Carlo Analysis in Academic Research

“…Wei & Tanner (1990) showed that MCMC simulation can be used to make the E step more practical. A number of similar approaches for the use of MCMC in the EM algorithm have been tested and found workable for particular classes of problems (Geyer, 1992a;Nielsen, 2000;Jank & Booth, 2003;Caffo et al, 2005;Marschner, 2001;Valpine, 2003). Second, very recent publications indicate that MCMC calculations can be used to derive ML estimates.…”

Nonlinear panel data estimation via quantile regressions

“…The sequence of parameter estimates converges to an ergodic Markov Chain in the limit. Following Nielsen (2000aNielsen ( , 2000b we characterize the asymptotic distribution of our sequential method-ofmoments estimators based on M imputations. A difference with most applications of EMtype algorithms is that we do not update parameters in each iteration using maximum likelihood, but using quantile regressions.…”