  The proposed approach extends the confidence posterior distribution to the
semi-parametric empirical Bayes setting. Whereas the Bayesian posterior is
defined in terms of a prior distribution conditional on the observed data, the
confidence posterior is defined such that the probability that the parameter
value lies in any fixed subset of parameter space, given the observed data, is
equal to the coverage rate of the corresponding confidence interval. A
confidence posterior that has correct frequentist coverage at each fixed
parameter value is combined with the estimated local false discovery rate to
yield a parameter distribution from which interval and point estimates are
derived within the framework of minimizing expected loss. The point estimates
exhibit suitable shrinkage toward the null hypothesis value, making them
practical for automatically ranking features in order of priority. The
corresponding confidence intervals are also shrunken and tend to be much
shorter than their fixed-parameter counterparts, as illustrated with gene
expression data. Further, simulations confirm a theoretical argument that the
shrunken confidence intervals cover the parameter at a higher-than-nominal
frequency.
