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

arXiv: https://arxiv.org/abs/2409.19812 · HTML: https://mlsanity.com/paper/2409.19812/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Sep 2024.

## Built on by (most cited first, 3 of 3)

- [Beyond Fixed False Discovery Rates: Post-Hoc Conformal Selection with E-Variables](https://mlsanity.com/paper/2604.11305/) (2026)
- [Bentkus-type asymptotic e-values](https://mlsanity.com/paper/2606.06332/) (2026)
- [Sequential Conditional Independence Testing with Machine Learning Models](https://mlsanity.com/paper/2610.11388/) (2026)

## Abstract (arXiv)

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. 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. As one example, we construct asymptotic compound e-values for multiple t-tests, where the (nuisance) variances may be different across hypotheses. Our construction may be interpreted as a data-driven instantiation of the optimal discovery procedure, and our results provide the first type-I error guarantees for the same, along with significant power gains.