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**Consistency of Bernstein Polynomial Posteriors**

**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 *P*_{0} 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.

*Heidi Sestrich*

*2/15/2000*
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