PREDICTIVE INFERENCE , RAREEVENTS AND HIERARCHICAL MODELSc Richard

Many problems of statistics are naturally formulated in terms of the predictive distribution of an as yet unobserved random variable, rather than the more traditional questions of parameter estimation and hypothesis testing. Both frequentist and Bayesian approaches to predictive inference may be considered, but it is argued that the Bayesian approach is more general and exible. This leads us to consider the frequentist properties of Bayesian procedures. A second-order asymptotic theory is developed, for comparing the risks of diierent procedures under a variety of loss functions. Particular attention is paid to the tails of the predictive distribution. This leads to some surprising but general conclusions. The usual Bayesian method may well be inferior to a crude maximum likelihood \plug-in" approach, but in most cases the comparison is reversed with a more general speciication of the Bayesian procedure. An extension is outlined to empirical Bayes problems. The classical James-Stein theorem is reinterpreted as an asymptotic result about the properties of Bayesian procedures, and it is shown how the same ideas may be applied to a general class of Bayesian hierarchical models.