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

Abstract

While empirical Bayes methods thrive in the presence of the 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 tested hypotheses. Indeed, for fewer hypotheses, there is more uncertainty in empirical Bayes estimates of the prior distribution. Confidence intervals 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. 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 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.

Keywords:

• Approximate confidence distribution
• bioequivalence
• bounded parameter
• empirical Bayes methods
• epistemic probability
• evidential model
• evidential support distribution
• fiducial model averaging
• restricted parameter space

Additional information

Funding

The comments of the anonymous reviewer led to a clearer presentation of the ideas. This research was partially supported by the Natural Sciences and Engineering Research Council of Canada (RGPIN/356018-2009). The major revisions were completed in the Ottawa office space generously provided by St. Paul Lutheran Church during the spring of 2020.