## Consistency of Bernstein Polynomial Posteriors

February, 2000

Tech Report

### Author(s)

Sonia Petrone and Larry Wasserman

### Abstract

A Bernstein prior is a probability measure on the space of all the distribution functions on [0,1]. Under very general assumptions, it selects absolutely continuous distribution functions, whose densities are mixtures of known beta densities. The Bernstein prior is of interest in Bayesian nonparametric inference with continuous data.

In this paper we study the consistency of the posterior from a Bernstein prior. We first show that, under mild assumptions, the posterior is weakly consistent for any distribution function P0 on [0,1] with continuous and bounded Lebesgue density.

With slightly stronger assumptions on the prior, the posterior is also Hellinger-consistent. This implies that the predictive density from a Bernstein prior, which is a Bayesian density estimate, converges in Hellinger sense to the true density (assuming it continuous and bounded). We also study a sieve maximum likelihood version of the density estimator, and show that it is also Hellinger consistent under weak assumption.

When the order of the Bernstein polynomial, i.e. the number of components in the beta-mixture, is truncated, we show that under mild restrictions, the posterior concentrates on the set of so called pseudo-true densities. Finally, we study the behavior of the predictive density numerically and we also study a hybrid Bayes-maximum likelihood density estimator.