Asymptotic and compound e-values: multiple testing and empirical Bayes

Understand

We explicitly define the notions of (bona fide, approximate or asymptotic) compound p-values and e-values, which have been implicitly presented and used in the recent multiple testing literature.

• While it is known that the e-BH procedure with compound e-values controls the FDR, we show the converse: 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 arbitrary FDR procedures.
• We then connect compound e-values to empirical Bayes.