The multivariate point null testing problem: A Bayesian discussion

In this paper the problem of testing a multivariate point hypothesis is considered. Of interest is the relationship between the p-value and the posterior probability. A Bayesian test for simple H0:θ=θ0 versus bilateral H0:θ≠θ0, with a mixed prior distribution for the parameter θ, is developed. The methodology consists of fixing a sphere of radius δ around θ0 and assigning a prior mass, π0, to H0 by integrating the density π(θ) over this sphere and spreading the remainder, 1−π0, over H1 according to π(θ). A theorem that shows when the frequentist and Bayesian procedures can give rise to the same decision is proved. Then, some examples are revisited.