The multivariate point null testing problem: A Bayesian discussion

Abstract

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 H 0 : θ = θ 0 versus bilateral H 0 : θ ≠ θ 0 , with a mixed prior distribution for the parameter θ , is developed. The methodology consists of fixing a sphere of radius δ around θ 0 and assigning a prior mass, π 0 , to H 0 by integrating the density π ( θ ) over this sphere and spreading the remainder, 1 − π 0 , over H 1 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.