On Bayesian Estimation of Densities and Sampling Distributions: the Posterior Predictive Distribution as the Bayes Estimator

TL;DR

This paper demonstrates that the posterior predictive distribution serves as the optimal Bayes estimator for density and sampling distribution estimation problems, with proofs of consistency and illustrative examples.

Contribution

It establishes the posterior predictive distribution as the Bayes estimator for specific density and sampling distribution estimation problems, including proofs of consistency.

Findings

Posterior predictive distribution is optimal for estimating sampling distributions.

The estimator of the density is shown to be consistent.

Examples illustrate the practical application of the theoretical results.

Abstract

Optimality results for two outstanding Bayesian estimation problems are given in this paper: the estimation of the sampling distribution for the squared total variation function and the estimation of the density for the $L^1$-squared loss function. The posterior predictive distribution provides the solution to these problems. Some examples are presented to illustrate it. The Bayesian estimation problem of a distribution function is also addressed. Consistency of the estimator of the density is proved.