  The following zero-sum game between nature and a statistician blends Bayesian
methods with frequentist methods such as p-values and confidence intervals.
Nature chooses a posterior distribution consistent with a set of possible
priors. At the same time, the statistician selects a parameter distribution for
inference with the goal of maximizing the minimum Kullback-Leibler information
gained over a confidence distribution or other benchmark distribution. An
application to testing a simple null hypothesis leads the statistician to
report a posterior probability of the hypothesis that is informed by both
Bayesian and frequentist methodology, each weighted according how well the
prior is known.
  Since neither the Bayesian approach nor the frequentist approach is entirely
satisfactory in situations involving partial knowledge of the prior
distribution, the proposed procedure reduces to a Bayesian method given
complete knowledge of the prior, to a frequentist method given complete
ignorance about the prior, and to a blend between the two methods given partial
knowledge of the prior. The blended approach resembles the Bayesian method
rather than the frequentist method to the precise extent that the prior is
known.
  The problem of testing a point null hypothesis illustrates the proposed
framework. The blended probability that the null hypothesis is true is equal to
the p-value or a lower bound of an unknown Bayesian posterior probability,
whichever is greater. Thus, given total ignorance represented by a lower bound
of 0, the p-value is used instead of any Bayesian posterior probability. At the
opposite extreme of a known prior, the p-value is ignored. In the intermediate
case, the possible Bayesian posterior probability that is closest to the
p-value is used for inference. Thus, both the Bayesian method and the
frequentist method influence the inferences made.
