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Bayes Factors, Nuisance Parameters and Imprecise Tests

Isabella Verdinelli and Larry Wasserman

Abstract:

We give conditions under which the Bayes test of a precise hypothesis approximates the test of an imprecise hypothesis in the presence of nuisance parameters. Specifically, if the likelihood is bounded and continuous, then it suffices that the prior under the null hypothesis be the generalized conditional of the prior under the alternative. The form of the conditional is determined by the sequence of approximating null hyptheses. If the likelihood is not continuous, it may not be possible to approximate an imprecise test with the precise test. When there are no nuisance parameters, the precise test approximates the imprecise test under much weaker conditions than is widely believed; in particular, it is not necessary to have sharply peaked priors.

KEY WORDS AND PHRASES: Bayes Factor, hypothesis testing, precise tests, imprecise tests.