Exponential posterior consistency via generalized Pólya urn schemes in finite semiparametric mixtures

Abstract

Advances in Markov chain Monte Carlo MCMC methods now make it computationally feasible and relatively straightforward to apply the Dirichlet process prior in a wide range of Bayesian nonparametric problems. The feasibility of these methods rests heavily on the fact that the MCMC approach avoids direct sampling of the Dirichlet process and is instead based on sampling the finite-dimensional posterior which is obtained from marginalizing out the process.

In application, it is the integrated posterior that is used in the Bayesian nonparametric inference, so one might wonder about its theoretical properties. This paper presents some results in this direction. In particular, we will focus on a study of the posterior’s asymptotic behavior, specifically for the problem when the data is obtained from a finite semiparametric mixture distribution. A complication in the analysis arises because the dimension for the posterior, although finite, increases with the sample size. The analysis will reveal general conditions that ensure exponential posterior consistency for a finite dimensional parameter and which can be slightly generalized to allow the unobserved nonparametric parameters to be sampled from a generalized Pólya urn scheme. Several interesting examples are considered.