Two Papers on the Comparison of Bayesian and Frequentist Approaches to Statistical Problems of Prediction: Bayesian Tolerance Regions

Abstract

In the theory of statistical tolerance regions, as usually presented in frequentist terms, there are inherent difficulties of formulation, development and interpretation. The present paper re-examines the basic problem from a Bayesian point of view and suggests that such an approach provides a set of widely applicable, mathematically tractable tools, often more tailored to the requirements of users than the corresponding frequentist tools. For the one-dimensional case, Bayesian intervals are quoted for a number of standard distributions and prior densities, and the customary feature of a Bayesian analysis—that special prior densities give rise to standard frequentist results—is briefly demonstrated. A problem which seems to be of greater practical significance, namely the selection of an optimum tolerance region from a set of possible tolerance regions, is also investigated and the overwhelming advantages of the Bayesian approach are indicated.