Inequalities for Bayes Factors and Relative Belief Ratios

We discuss the definition of a Bayes factor, the Savage-Dickey result, and develop some inequalities relevant to Bayesian inferences. We consider the implications of these inequalities for the Bayes factor approach to hypothesis assessment. An approach to hypothesis assessment based on the computation of a Bayes factor, a measure of reliability of the Bayes factor, and the point where the Bayes factor is maximized is recommended. This can be seen to deal with many of the issues and controversies associated with hypothesis assessment. It is noted that an inconsistency in prior assignments can arise when priors are placed on hypotheses that do not arise from a parameter of interest. It is recommended that this inconsistency be avoided by choosing a distance measure from the hypothesis as the parameter of interest. An application is made to assessing the goodness of fit for a logistic regression model and it is shown that this leads to resolving some diffi culties associated with assigning priors for this model.