Importance of the prior mass for agreement between frequentist and Bayesian approaches in the two-sided test

A Bayesian test for H-0: = (0) versus H-1: (0) is developed. The methodology consists of fixing a sphere of radius around (0), assigning to H-0 a prior mass, (0), computed by integrating a density function () over this sphere, and spreading the remainder, 1(0), over H-1 according to (). The ultimate goal is to show when p values and posterior probabilities can give rise to the same decision in the following sense. For a fixed level of significance , when do (12) exist such that, regardless of the data, a Bayesian proponent who uses the proposed mixed prior with (0)((1), (2)) reaches, by comparing the posterior probability of H-0 with 1/2, the same conclusion as a frequentist who uses to quantify the p value? A theorem that provides the required constructions of (1) and (2) under verification of a sufficient condition ((12)) is proved. Some examples are revisited.