On The Justification of Default and Intrinsic Bayes Factors

In Bayesian model selection or hypothesis testing, it is difficult to develop default Bayes factors, since (improper) noninformative priors cannot typically be used. In developing such default Bayes factors, we feel that it is important to keep several principles in mind. The first is that the default Bayes factor should correspond, in some sense, to an actual Bayes factor with a (sensible) prior, which we call an intrinsic prior. The second principle is that such priors should be properly calibrated across models, in the sense of being “predictively matched.” These notions will be described and illustrated, primarily using examples involving the intrinsic Bayes factor, a recently proposed default Bayes factor. It will be seen that intrinsic Bayes factors seem to correspond to actual Bayes factors with proper priors, at least for nested model scenarios. The corresponding intrinsic priors are specifically given for the normal linear model.