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How to be a better Bayesian. (English) Zbl 0869.62006
Summary: Consider an experiment yielding an observable random quantity $X$ whose distribution is indexed by a real parameter $\theta$. Suppose that a statistician is prepared to execute a Bayes procedure in estimating $\theta$ and has quantified available prior information about $\theta$ 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. We investigate the questions of whether, and when, this additional information can be exploited so as to provide a better estimate of $\theta$ 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 $\theta$ 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 combining some relevant empirical findings with the subjective information embodied in the prior $G$. Our results answer a question in a Bayesian context that remains exclusive in the classical, non-Bayesian version of empirical Bayes estimation. In the final section, our BEB estimator is shown to compare favorably to two other BEB estimators suggested in the literature.

##### MSC:
 62A01 Foundations and philosophical topics in statistics 62C12 Empirical decision procedures; empirical Bayes procedures
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