Consistency of Variational Bayes Inference for Estimation and Model Selection in Mixtures

TL;DR

This paper investigates the theoretical properties of variational Bayes inference in mixture models, establishing consistency, convergence rates, and validating model selection criteria through rigorous proofs and applications.

Contribution

It provides the first theoretical analysis of variational approximations in mixture models, including consistency, convergence rates, and validation of the evidence lower bound for model selection.

Findings

Variational Bayes approximations are consistent for mixture models.

The convergence rates of variational posteriors are established.

Maximizing the evidence lower bound yields strong oracle inequalities.

Abstract

Mixture models are widely used in Bayesian statistics and machine learning, in particular in computational biology, natural language processing and many other fields. Variational inference, a technique for approximating intractable posteriors thanks to optimization algorithms, is extremely popular in practice when dealing with complex models such as mixtures. The contribution of this paper is two-fold. First, we study the concentration of variational approximations of posteriors, which is still an open problem for general mixtures, and we derive consistency and rates of convergence. We also tackle the problem of model selection for the number of components: we study the approach already used in practice, which consists in maximizing a numerical criterion (the Evidence Lower Bound). We prove that this strategy indeed leads to strong oracle inequalities. We illustrate our theoretical results by applications to Gaussian and multinomial mixtures.