Bayesian and Frequentist Approaches to Parametric Predictive Inference

SUMMARY Suppose we want to estimate predictive probabilities for an unobserved random variable, whose distribution depends on an unknown nite-dimensional parameter which is estimated from past data. A crude \plug-in" approach is to substitute a point estimate of the unknown parameter into the predictive distribution. This may be criticized as failing to take into account the uncertainty of the unknown parameter, whereas a Bayesian approach incorporates such uncertainty in a natural way. Therefore, we might expect that a Bayesian procedure will be superior to a plug-in approach even when assessed in purely frequentist terms. The analysis given in this paper, however, shows that this is too simple a conclusion. For many models, when assessed from the point of view of mean squared error of predictive probabilities, the plug-in approach is superior to the Bayesian approach in the extreme tail of the distribution. We may also consider alternative loss functions besides mean squared error, but in most cases a similar phenomenon occurs. However, one may also generalize the class of Bayesian procedures by considering alternatives to the posterior mean of the predictive distribution. With such generalizations, the comparison swings back in favor of Bayesian procedures. These results are initially developed for simple models, in particular the exponential distribution, but in later discussion they are used to study the properties of hierarchical modeling approaches to empirical Bayes problems. The mathematical development is only sketched in the present paper, but relies heavily on asymptotic expansions and second-order properties.