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

Abstract

Summary 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.