Empirical partially Bayes multiple testing and compound χ² decisions

We study multiple testing in the normal means problem with estimated
variances that are shrunk through empirical Bayes methods. The situation is asymmetric in
that a prior is posited for the nuisance parameters (variances) but not the primary
parameters (means).
If the prior were known, one could proceed by computing p-values
conditional on sample variances; a strategy called partially Bayes inference by Sir David
Cox. These conditional p-values satisfy a Tweedie-type formula and are approximated at
nearly-parametric rates when the prior is estimated by nonparametric maximum likelihood. If
the variances are in fact fixed, the approach retains type-I error guarantees. As is common
in the empirical Bayes paradigm, our results hinge on the interpretation of the prior as the
frequency distribution of the nuisance parameters, and should be contrasted with e.g., the
conditional predictive p-values of Bayarri and Berger.

Based on joint work with Bodhisattva Sen.