Compound e-values and empirical Bayes

We explicitly define the notion of (exact or approximate) compound e-values which have been implicitly presented and extensively used in the recent multiple testing literature. We show that every FDR controlling procedure can be recovered by instantiating the e-BH procedure with certain compound e-values. Since compound e-values are closed under averaging, this allows for combination and derandomization of any FDR procedure. We then point out and exploit the connections to empirical Bayes. In particular, we use the fundamental theorem of compound decision theory to derive the log-optimal simple separable compound e-value for testing a set of point nulls against point alternatives: it is a ratio of mixture likelihoods. We extend universal inference to the compound setting. As one example, we construct approximate compound e-values for multiple t-tests, where the (nuisance) variances may be different across hypotheses. We provide connections to related notions in the literature stated in terms of p-values.