Sensitivity of the fractional Bayes factor to prior distributions

Calculation of a suitable Bayes factor is required for Bayesian model comparison. The fractional Bayes factor is one of several alternative Bayes factors that have been introduced in recent years to address the problem of sensitivity of the usual Bayes factor when prior information is weak. Sensitivity of the fractional Bayes factor with respect to prior distributions is easy to assess when these are proper. On the other hand, when the priors are improper, most methods lead to trivial answers. Also, earlier work on fractional Bayes factors has assumed that sensitivity will be reduced if the training fraction, b, is increased, but this has only been justified by appeal to heuristic reasoning and simple examples. In this paper we derive a measure of the sensitivity of the fractional Bayes factor with respect to improper priors, and prove that it is a decreasing function of b in a class of problems.