The Consistency of Posterior Distribtions in Nonparametric Problems

Andrew Barron, Mark J. Schervish and Larry Wasserman


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.

Keywords: Exponential families, Hellinger distance, Nonparametric Bayesian inference, Polya trees.

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