The Whetstone and the Alum Block: Balanced Objective Bayesian Comparison of Nested Models for Discrete Data

When two nested models are compared, using a Bayes factor, from an objective standpoint, two seemingly conflicting issues emerge at the time of choosing parameter priors under the two models. On the one hand, for moderate sample sizes, the evidence in favor of the smaller model can be inflated by diffuseness of the prior under the larger model. On the other hand, asymptotically, the evidence in favor of the smaller model typically accumulates at a slower rate. With reference to finitely discrete data models, we show that these two issues can be dealt with jointly, by combining intrinsic priors and nonlocal priors in a new unified class of priors. We illustrate our ideas in a running Bernoulli example, then we apply them to test the equality of two proportions, and finally we deal with the more general case of logistic regression models.