On the Consistency of Bayes Factors for Testing Point Null versus Nonparametric Alternatives

When testing a point null hypothesis versus an alternative that is vaguely speciied, a Bayesian test usually proceeds by putting a nonparametric prior on the alternative and then computing a Bayes factor based on the observations. This paper addresses the question of consistency, that is, whether the Bayes factor is correctly indicative of the null or the alternative as sample size increases. We establish several consistency results in the aarmative under fairly general conditions. Consistency of Bayes factors for testing a point null versus a parametric alternative has long been known. The results here can also be viewed as the nonparametric extension of their parametric counterpart.