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The following zero-sum game between nature and a statistician blends Bayesianmethods with frequentist methods such as p-values and confidence intervals.Nature chooses a posterior distribution consistent with a set of possiblepriors. At the same time, the statistician selects a parameter distribution forinference with the goal of maximizing the minimum Kullback-Leibler informationgained over a confidence distribution or other benchmark distribution. Anapplication to testing a simple null hypothesis leads the statistician toreport a posterior probability of the hypothesis that is informed by bothBayesian and frequentist methodology, each weighted according how well theprior is known.  Since neither the Bayesian approach nor the frequentist approach is entirelysatisfactory in situations involving partial knowledge of the priordistribution, the proposed procedure reduces to a Bayesian method givencomplete knowledge of the prior, to a frequentist method given completeignorance about the prior, and to a blend between the two methods given partialknowledge of the prior. The blended approach resembles the Bayesian methodrather than the frequentist method to the precise extent that the prior isknown.  The problem of testing a point null hypothesis illustrates the proposedframework. The blended probability that the null hypothesis is true is equal tothe p-value or a lower bound of an unknown Bayesian posterior probability,whichever is greater. Thus, given total ignorance represented by a lower boundof 0, the p-value is used instead of any Bayesian posterior probability. At theopposite extreme of a known prior, the p-value is ignored. In the intermediatecase, the possible Bayesian posterior probability that is closest to thep-value is used for inference. Thus, both the Bayesian method and thefrequentist method influence the inferences made.

Blending Bayesian and frequentist methods according to the precision of prior information with an application to hypothesis testing