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Asymptotic and compound e-values: multiple testing and empirical Bayes
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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.
Forward citations
Cited by 4 Pith papers
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False discovery rate control with compound p-values
The paper derives finite-sample bounds on the false discovery rate of the Benjamini-Hochberg procedure for compound p-values: a constant-factor bound under independence, a near-optimal quadratic bound under the global...
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The e-Partitioning Principle of False Discovery Rate Control
The e-Partitioning Principle states that a procedure controls FDR if and only if it is a subset of a general e-value based partitioning procedure, yielding uniform improvements of eBH, BY and Su.
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Bringing Closure to False Discovery Rate Control: A General Principle for Multiple Testing
Every multiple testing method that controls an expected loss such as FDR is a special case of a single e-value based closed testing procedure, and the closed versions of eBH, BY, and Su reject at least as much.
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Anytime-valid FDR control with the stopped e-BH procedure
Stopped e-BH controls FDR at all stopping times when the underlying e-processes are global, and local e-processes become global under a no-unobserved-confounding Markov condition.
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