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The Consistency of Posterior Distribtions in Nonparametric Problems

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.

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



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