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:[theta]=[theta]0 versus bilateral H0:[theta][not equal to][theta]0, with a mixed prior distribution for the parameter [theta], is developed. The methodology consists of fixing a sphere of radius [delta] around [theta]0 and assigning a prior mass, [pi]0, to H0 by integrating the density [pi]([theta]) over this sphere and spreading the remainder, 1-[pi]0, over H1 according to [pi]([theta]). A theorem that shows when the frequentist and Bayesian procedures can give rise to the same decision is proved. Then, some examples are revisited.