STOR Colloquium: Subhashis Ghosal (North Carolina State University)

Adaptive Bayesian Multivariate Density Estimation with Dirichlet Mixtures

The kernel method has been an extremely important component in the nonparametric toolbox. It has undergone tremendous development since its introduction over fifty years ago. Bayesian methods for density estimation using kernel-smoothed priors were first introduced in the mid-eighties, where a random probability measure following typically a Dirichlet process is convoluted with a kernel to induce a prior on smooth densities. The resulting prior distribution is commonly known as a Dirichlet mixture process. Such priors gained popularity in the Bayesian nonparametric literature after the development of Markov chain Monte-Carlo methods for posterior computation. Posterior consistency of a Dirichlet mixture prior with a normal kernel was established by Ghosal et al. (1999, Annals of Statistics). Subsequent papers relaxed conditions for consistency, generalized to other kernels and studied rates of convergence, especially in the univariate case. More recently, it has been found that Bayesian kernel mixtures of finitely supported random distributions have some automatic rate adaptation property --- something a classical kernel estimator lacks. We consider Bayesian multivariate density estimation using a Dirichlet mixture of normal kernel as the prior distribution. By representing a Dirichlet process as a stick-breaking process, we are able to extend convergence results beyond finitely supported mixtures priors to Dirichlet mixtures. Thus our results have new implications in the univariate situation as well. Assuming that the true density satisfies Holder smoothness and exponential tail conditions, we show that the rates of posterior convergence are minimax-optimal up to a logarithmic factor. This procedure is fully adaptive since the priors are constructed without using the knowledge of the smoothness level.

This is a joint work with Weining Shen, a graduate student at Department of Statistics in North Carolina State University.

Refreshments will be served at 3:30pm in the 3rd floor lounge of Hanes Hall