## The Consistency of Posterior Distribtions in Nonparametric Problems

January, 1997

Tech Report

### Author(s)

Andrew Barron, Mark J. Schervish and Larry Wasserman

### Abstract

We give conditions that guarantee that the posterior probability of every Hellinger neighborhood of the true density tends to 1 almost surely. The conditions are (i) a smoothness condition on the prior and (ii) a requirement that the prior put positive mass in appropriate neighborhoods of the true density. The results are based on the idea of approximating the set of densities with a finite dimensional set of densities and then computing the Hellinger bracketing metric entropy of the approximating set. We apply the results to some examples.