10 - Bayes Procedures

Summary

In this chapter Bayes estimators are studied from a frequentist perspective. Both posterior measures and Bayes point estimators in smooth parametric models are shown to be asymptotically normal.

Introduction

In Bayesian terminology the distribution of an observation in under a parameteris viewed as the conditional law of in given that a random variable en is equal to. The distribution n of the “random parameter” en is called the prior distribution, and the conditional distribution of en given in is the posterior distribution. If en possesses a density and admits a density (relative to given dominating measures), then the density of the posterior distribution is given by Bayes’ formula

This expression may define a probability density even ifis not a probability density itself. A prior distribution with infinite mass is called improper.

The calculation of the posterior measure can be considered the ultimate aim of a Bayesian analysis. Alternatively, one may wish to obtain a “point estimator” for the parameter using the posterior distribution. The posterior mean is often used for this purpose, but other location estimators are also reasonable.

A choice of point estimator may be motivated by a loss function. The Bayes risk of an estimator T_n relative to the loss function i and prior measure n is defined as

Here the expectation is the risk function of T_n in the usual set-up and is identical to the conditional risk in the Bayesian notation. The corresponding Bayes estimator is the estimator T_n that minimizes the Bayes risk. Because the Bayes risk can be written in the the value minimizes, for every fixed the “posterior risk”

Minimizing this expression may again be a well-defined problem even for prior densities of infinite total mass. For the loss function, the solution T_n is the posterior mean, the solution is the posterior median.