A Note on Consistency of the Bayes Estimator of the Density

TL;DR

This paper proves the strong consistency of the Bayes density estimator under mild conditions and shows its Bayes risk approaches zero as sample size increases, also applying similar results to sampling distribution estimation.

Contribution

It establishes the strong consistency and asymptotic optimality of the Bayes density estimator and posterior predictive density under mild assumptions.

Findings

Bayes density estimator is strongly consistent.

Bayes risk of the estimator approaches zero with increasing sample size.

Results extend to estimation of the sampling distribution.

Abstract

Under mild conditions, it is shown the strong consistency of the Bayes estimator of the density. Moreover, the Bayes risk (for some common loss functions) of the Bayes estimator of the density (i.e. the posterior predictive density) reaches zero when the sample size goes to $\infty$. In passing, a similar result is obtained for the estimation of the sampling distribution.