Objective Bayes and Conditional Frequentist Inference

Abstract

Objective Bayesian methods have garnered considerable interest and support among statisticians, particularly over the past two decades. It has often been ignored, however, that in some cases the appropriate frequentist inference to match is a conditional one. We present various methods for extending the probability matching prior (PMP) methods to conditional settings. A method based on saddlepoint approximations is found to be the most tractable and we demonstrate its use in the most common exact ancillary statistic models. As part of this analysis, we give a proof of an exactness property of a particular PMP in location-scale models. We use the proposed matching methods to investigate the relationships between conditional and unconditional PMPs. A key component of our analysis is a numerical study of the performance of probability matching priors from both a conditional and unconditional perspective in exact ancillary models. In concluding remarks we propose many routes for future research.