Assessing post-data weight of evidence

Under a robust Bayesian framework, we study p-values as post-data weights of evidence in hypothesis testing problems. For one-sided and two-sided hypothesis testing problems for the normal mean, p-values are considered as maximum likelihood estimates for some functions of the mean. We show that in contrast to Bayes estimates for reasonable families of priors, p-values are extreme for one-sided hypothesis testing problems and are moderate for two-sided problems. Implications on the controversies of ir/reconcilability of p-values in Casella and Berger [1987, J. Am. Statist. Assoc. 82, 106-111], Berger and Sellke [1987, J. Am. Statist. Assoc. 82, 112-122] and Berger and Delampady [1987, Statist. Sci. 2, 317-352] will also be addressed.