Conditions for convergence of Monte Carlo EM sequences with an application to product diffusion modeling

Abstract

Intractable maximum likelihood problems can sometimes be finessed with a Monte Carlo implementation of the E M algorithm. However, there appears to be little theory governing when Monte Carlo E M (MC E M) sequences converge. Consequently, in some applications, convergence is assumed rather than proved. Motivated by this problem in the context of modeling market penetration of new products and services over time, we develop (i) high-level conditions for rates of almost-sure convergence and convergence in distribution of any MC E M sequence and (ii) primitive conditions for almost-sure monotonicity and almostsure convergence of an MC E M sequence when Monte Carlo integration is carried out using independent Gibbs runs. We verify the main primitive conditions for the Bass product diffusion model and apply the methodology to data on wireless telecommunication services.

Language Evolution by Iterated Learning With Bayesian Agents

“…Statisticians have explored a great many variants on the EM algorithm, some of which have natural interpretations in the context of iterated learning. In particular, Monte Carlo EM algorithms where m n > 1 (as opposed to stochastic EM, where m n = 1) are directly applicable to cases of iterated learning, and have been studied extensively (Sherman, et al, 1999;Fort & Moulines, 203). Statisticians have also investigated a version of the stochastic EM algorithm in which the samples of latent variables from previous iterations are also incorporated, providing a natural way of modeling language evolution when learners are exposed to linguistic data produced by more than one previous generation (Celeux & Diebolt, 1992;Celeux et al, 1995;Delyon, Lavielle, & Moulines, 1999).…”

Exact and Computationally Efficient Likelihood-Based Estimation for Discretely Observed Diffusion Processes (with Discussion)

“…However, since we can simulate U , and X u for any u by using the EA, we adopt an MC implementation of the EM algorithm. The MCEM algorithm was introduced in Wei and Tanner (1990); convergence and implementation issues were tackled in Chan and Ledolter (1995), Sherman et al. (1999) and Fort and Moulines (2003).…”

“…However, since we can simulate U, and X u for any u by using the EA, we adopt an MC implementation of the EM algorithm. The MCEM algorithm was introduced in Wei and Tanner (1990); convergence and implementation issues were tackled in Chan and Ledolter (1995), Sherman et al (1999) and Fort and Moulines (2003). It is well documented (see for example Fort and Moulines (2003), and references therein) that the number of MC samples that are used to approximate the expectation should increase with the EM iterations.…”

Joint analysis of occurrence and time to stability after entrance into the Italian labour market: an approach based on a Bayesian cure model with structured stochastic search variable selection

“…When random intercepts are included in order to take into account unobserved heterogeneity, the EM algorithm shall be modified to an Monte Carlo expectation-maximization algorithm (e.g. [40,42]). These algorithms are described in Appendix 1.…”