Bayes Factors, Nuisance Parameters and Imprecise Tests
Isabella Verdinelli, Larry Wasserman · 1996
Abstract Bayes factors are often used to test precise null hypotheses. One objection to the Bayes factor B is that a precise null hypothesis can sometimes be unrealistic. But it can be shown that B approximates the Bayes factor for an imprecise null hypothesis, under very mild conditions. We show that a similar approximation holds when there are nuisance parameters. Specifically, if the likelihood is bounded and continuous, then it suffices that the prior under the null hypothesis be the geometric conditional of the prior under the alternative. The form of the conditional is determined by the sequence of approximating null hypotheses. If the likelihood is not continuous, it may not be possible to approximate an imprecise test with the precise test.