How to be a Better Bayesian

Abstract

Abstract Consider an experiment yielding an observable random quantity X whose distribution is indexed by a real parameter o. Suppose that a statistician is prepared to execute a Bayes procedure in estimating o and has quantified available prior information about o in his chosen prior distribution G. Suppose that before this estimation process is completed, the statistician becomes aware of the outcome Y of a “similar” experiment. In this article we investigate the questions of whether, and when, this additional information can be exploited so as to provide a better estimate of o in the “current” experiment. We show that in the traditional empirical Bayes framework and in situations involving exponential families, conjugate priors, and squared error loss, the answer is “essentially always.” An explicit Bayes empirical Bayes (BEB) estimator of o is given that is superior to the original Bayes estimator, showing that the statistician has, in these problems, the opportunity to be a “better Bayesian” by combi...