On consistency of nonparametric normal mixtures for Bayesian density estimation

The past decade has seen a remarkable development in the area of Bayesian nonparametric inference from both theoretical and applied perspectives. As for the latter, the celebrated Dirichlet process has been successfully exploited within Bayesian mixture models, leading to many interesting applications. As for the former, some new discrete nonparametric priors have been recently proposed in the literature that have natural use as alternatives to the Dirichlet process in a Bayesian hierarchical model for density estimation. When using such models for concrete applications, an investigation of their statistical properties is mandatory. Of these properties, a prominent role is to be assigned to consistency. Indeed, strong consistency of Bayesian nonparametric procedures for density estimation has been the focus of a considerable amount of research; in particular, much attention has been devoted to the normal mixture of Dirichlet process. In this article we improve on previous contributions by establishing strong consistency of the mixture of Dirichlet process under fairly general conditions. Besides the usual Kullback–Leibler support condition, consistency is achieved by finiteness of the mean of the base measure of the Dirichlet process and an exponential decay of the prior on the standard deviation. We show that the same conditions are also sufficient for mixtures based on priors more general than the Dirichlet process. This leads to the easy establishment of consistency for many recently proposed mixture models.