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

Abstract

For the one-sided hypothesis testing problem it is shown that it is possible to reconcile Bayesian evidence against H ₀, expressed in terms of the posterior probability that H ₀ is true, with frequentist evidence against H ₀, expressed in terms of the p value. In fact, for many classes of prior distributions it is shown that the infimum of the Bayesian posterior probability of H ₀ is equal to the p value; in other cases the infimum is less than the p value. The results are in contrast to recent work of Berger and Sellke (1987) in the two-sided (point null) case, where it was found that the p value is much smaller than the Bayesian infimum. Some comments on the point null problem are also given.

Key Words:

• Posterior probability
• p Value
• Prior distribution