Bayesian Nonparametric Mixture Models

Abstract

This chapter reviews some of the popular Bayesian nonparametric mixture models, starting with the widely used Dirichlet process (DP) mixture models. It provides some generalizations of the DP prior and discusses inference that exploits the posterior distribution on the implied random partition and variations of these models. The chapter also reviews the use of repulsive priors on the mixing measure, using in particular the determinantal point process (DPP). It explores the use of Bayesian nonparametric priors for inference in mixtures and explains some commonly used models, including in particular the Dirichlet process prior, normalized random measures with independent increments, and the DPP and variations. Interpreting a mixture model as an expectation with respect to a mixing measure, it becomes natural to complete the model with a prior probability model on the unknown mixing measure. Inference for mixture models and closely related hierarchical models is one of the big success stories of Bayesian inference.