Hypothesis Assessment Using the Bayes Factor and Relative Belief Ratio

The Bayes factor is commonly used for assessing the evidence for or against a given hypothesis H0 : θ ∈ Θ0, where Θ0 is a subset of the parameter space. In this paper we discuss the Bayes factor and various issues associated with its use. A Bayes factor is seen to be intimately connected with a relative belief ratio which provides a somewhat simpler approach to assessing the evidence in favor of H0. It is noted that, when there is a parameter of interest generating H0, then a Bayes factor for H0 can be defined as a limit and there is no need to introduce a discrete prior mass for Θ0 or a prior within Θ0. It is further noted that when a prior on Θ0 does not correspond to a conditional prior induced by a parameter of interest generating H0, then there is an inconsistency in prior assignments. This inconsistency can be avoided by choosing a parameter of interest that generates the hypothesis. A natural choice of a parameter of interest is given by a measure of distance of the model parameter from Θ0. This leads to a Bayes factor for H0 that is comparing the concentration of the posterior about Θ0 with the concentration of the prior about Θ0. The issue of calibrating a Bayes factor is also discussed and is seen to be equivalent to computing a posterior probability that measures the reliability of the evidence provided by the Bayes factor.