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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.