Convergence of posterior odds

The problem of approximating an interval null or imprecise hypothesis test by a point null or precise hypothesis test under a Bayesian framework is considered. In the literature, some of the methods for solving this problem have used the Bayes factor for testing a point null and justified it as an approximation to the interval null. However, many authors recommend evaluating tests through the posterior odds, a Bayesian measure of evidence against the null hypothesis. It is of interest then to determine whether similar results hold when using the posterior odds as the primary measure of evidence. For the prior distributions under which the approximation holds with respect to the Bayes factor, it is shown that the posterior odds for testing the point null hypothesis does not approximate the posterior odds for testing the interval null hypothesis. In fact, in order to obtain convergence of the posterior odds, a number of restrictive conditions need to be placed on the prior structure. Furthermore, under a non-symmetrical prior setup, neither the Bayes factor nor the posterior odds for testing the imprecise hypothesis converges to the Bayes factor or posterior odds respectively for testing the precise hypothesis. To rectify this dilemma, it is shown that constraints need to be placed on the priors. In both situations, the class of priors constructed to ensure convergence of the posterior odds are not practically useful, thus questioning, from a Bayesian perspective, the appropriateness of point null testing in a problem better represented by an interval null. The theories developed are also applied to an epidemiological data set from White et al. (Can. Veterinary J. 30 (1989) 147–149.) in order to illustrate and study priors for which the point null hypothesis test approximates the interval null hypothesis test. AMS Classification: Primary 62F15; Secondary 62A15