  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.
