Empirical Bayes Interval Estimates that are Conditionally Equal to Unadjusted Confidence Intervals or to Default Prior Credibility Intervals

Abstract

Problems involving thousands of null hypotheses have been addressed by estimating the local false discovery rate (LFDR). A previous LFDR approach to reporting point and interval estimates of an effect-size parameter uses an estimate of the prior distribution of the parameter conditional on the alternative hypothesis. That estimated prior is often unreliable, and yet strongly influences the posterior intervals and point estimates, causing the posterior intervals to differ from fixed-parameter confidence intervals, even for arbitrarily small estimates of the LFDR. That influence of the estimated prior manifests the failure of the conditional posterior intervals, given the truth of the alternative hypothesis, to match the confidence intervals. Those problems are overcome by changing the posterior distribution conditional on the alternative hypothesis from a Bayesian posterior to a confidence posterior. Unlike the Bayesian posterior, the confidence posterior equates the posterior probability that the parameter lies in a fixed interval with the coverage rate of the coinciding confidence interval. The resulting confidence-Bayes hybrid posterior supplies interval and point estimates that shrink toward the null hypothesis value. The confidence intervals tend to be much shorter than their fixed-parameter counterparts, as illustrated with gene expression data. Simulations nonetheless confirm that the shrunken confidence intervals cover the parameter more frequently than stated. Generally applicable sufficient conditions for correct coverage are given. In addition to having those frequentist properties, the hybrid posterior can also be motivated from an objective Bayesian perspective by requiring coherence with some default prior conditional on the alternative hypothesis. That requirement generates a new class of approximate posteriors that supplement Bayes factors modified for improper priors and that dampen the influence of proper priors on the credibility intervals. While that class of posteriors intersects the class of confidence-Bayes posteriors, neither class is a subset of the other. In short, two first principles generate both classes of posteriors: a coherence principle and a relevance principle. The coherence principle requires that all effect size estimates comply with the same probability distribution. The relevance principle means effect size estimates given the truth of an alternative hypothesis cannot depend on whether that truth was known prior to observing the data or whether it was learned from the data.