On Bayesian Analysis of Mixtures with an Unknown Number of Components

This article is a contribution to the methodology of fully Bayesian mixture modelling. We stress the word "fully" in two senses. First, we model the number of components and the mixture component parameters jointly and base inference about these quantities on their posterior probabilities. This is in contrast to most previous Bayesian treatments of mixture estimation, which consider models for different numbers of components separately, and use significance tests or other non-Bayesian criteria to infer the number of components. Secondly, we aim to present posterior distributions of our objects of inference (model parameters and predictive densities), and not just "best estimates". There are three key ideas in our treatment. First, we demonstrate that novel MCMC methods, the "reversible jump" samplers introduced by Green (1994, 1995), can be used to sample mixture representations with an unknown and hence varying number of components. We believe these methods are preferable on grounds of convenience,