Bayesian and frequentist testing of the hypothesis \(H_ 0: \theta \leq 0\) against \(H_ 1: \theta >0\), where \(\theta\) is an unknown parameter, are considered. The p-value from the frequentist inference is compared with the posterior probability that \(H_ 0\) is true given the data where the prior is from classes of priors which are defined as impartial i.e. that which give equal weight to both the null and alternative hypotheses. For many of these classes it is shown that the infimum of this posterior probability is equal to p and other cases less than p. It is shown by J. O. Berger and T. Sellke [see the following review, Zbl 0612.62022] that these inferences are irreconcilable if \(H_ 0\) is a point hypothesis.
Discussion of both papers appears after the paper of Berger and Sellke.