Consistency of mixture models with a prior on the number of components

TL;DR

This paper provides general conditions ensuring posterior consistency in Bayesian finite mixture models with a prior on the number of components, guaranteeing convergence of the estimated parameters to the true values under broad conditions.

Contribution

It establishes almost sure posterior consistency for the number of components, weights, and parameters in finite mixture models using Doob's theorem under general conditions.

Findings

Posterior concentrates on true parameters with probability one.

Consistency holds for the number of components, weights, and parameters.

Results apply to common prior choices, ensuring practical relevance.

Abstract

This article establishes general conditions for posterior consistency of Bayesian finite mixture models with a prior on the number of components. That is, we provide sufficient conditions under which the posterior concentrates on neighborhoods of the true parameter values when the data are generated from a finite mixture over the assumed family of component distributions. Specifically, we establish almost sure consistency for the number of components, the mixture weights, and the component parameters, up to a permutation of the component labels. The approach taken here is based on Doob's theorem, which has the advantage of holding under extraordinarily general conditions, and the disadvantage of only guaranteeing consistency at a set of parameter values that has probability one under the prior. However, we show that in fact, for commonly used choices of prior, this yields consistency at Lebesgue-almost all parameter values -- which is satisfactory for most practical purposes. We aim to formulate the results in a way that maximizes clarity, generality, and ease of use.