Reconciling Bayesian and Frequentist Evidence in the One-Sided Scale Parameter Testing Problem

Abstract

An interesting but controversial problem arises when Bayesian as well as frequentist methodologies very often suggest similar solutions, thus creating a problem for the experimenter who wishes to make the best possible decision. For over a decade, there has been an effort by several authors to assess when Bayesian and frequentist methods provide exactly the same answers when employed. We encounter this situation in the problem of hypothesis testing, where Bayesian evidence, such as Bayes factors and prior or posterior predictive p-values are set against the classical p-values. All these measures provide a basis for rejection of a hypothesis or a model, but the selection of one over the other never comes without severe criticism. In this article, we develop prior predictive and posterior predictive p-values for one sided hypothesis testing for scale parameter problems. We reconcile Bayesian and frequentist evidence by showing that for many classes of prior distributions, the infimum of the prior predictive and posterior predictive p-values are equal to the classical p-value, for very general classes of distributions. The results are illustrated through examples relating to the one-sided testing problem for scale parameter.