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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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