Confidence distributions and empirical Bayes posterior distributions unified as distributions of evidential support

While empirical Bayes methods thrive in the presence of the hundreds or thousands of simultaneous hypothesis tests in genomics and other large-scale applications, significance tests and confidence intervals are considered more appropriate for small numbers of simultaneously tested hypotheses. Indeed, for fewer hypotheses or, more generally, fewer populations, there is more uncertainty in empirical Bayes methods of estimating the prior distribution. Confidence distributions have been used to propagate the uncertainty in the prior to empirical Bayes inference about a parameter, but only by combining a Bayesian posterior distribution with a confidence distribution, a probability distribution that encodes significance tests and confidence intervals. Combining distributions of both types has also been used to combine empirical Bayes methods and confidence intervals for estimating a parameter of interest. To clarify the foundational status of such combinations, the concept of an evidential model is proposed. In the framework of evidential models, both Bayesian posterior distributions and confidence distributions are degenerate special cases of evidential support distributions. Evidential support distributions, by quantifying the sufficiency of the data as evidence, leverage the strengths of Bayesian posterior distributions and confidence distributions for cases in which each type performs well and for cases benefiting from the combination of both. Evidential support distributions also address problems of bioequivalence, bounded parameters, and the lack of a unique confidence distribution.