Default Bayes Factors for Nonnested Hypothesis Testing

Abstract

Bayesian hypothesis testing for nonnested hypotheses is studied, using various “default” Bayes factors, such as the fractional Bayes factor, the median intrinsic Bayes factor, and the encompassing and expected intrinsic Bayes factors. The different default methods are first compared with each other and with the p value in normal one-sided testing, to illustrate the basic issues. General results for one-sided testing in location and scale models are then presented. The default Bayes factors are also studied for specific models involving multiple hypotheses. In particular, a multiple hypothesis testing example involving a sequential clinical trial is discussed. In most of the examples presented we also derive the intrinsic prior; this is the prior distribution, which, if used directly, would yield answers (asymptotically) equivalent to those for the given default Bayes factor.