Empirical partially Bayes multiple testing and compound $\chi^2$ decisions

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Publication date: 5 March 2023

Abstract: 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.

Has companion code repository: https://github.com/nignatiadis/empirical-partially-bayes-paper

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