False discovery rate control with e‐values

E‐values have gained attention as potential alternatives to p‐values as measures of uncertainty, significance and evidence. In brief, e‐values are realized by random variables with expectation at most one under the null; examples include betting scores, (point null) Bayes factors, likelihood ratios and stopped supermartingales. We design a natural analogue of the Benjamini‐Hochberg (BH) procedure for false discovery rate (FDR) control that utilizes e‐values, called the e‐BH procedure, and compare it with the standard procedure for p‐values. One of our central results is that, unlike the usual BH procedure, the e‐BH procedure controls the FDR at the desired level—with no correction—for any dependence structure between the e‐values. We illustrate that the new procedure is convenient in various settings of complicated dependence, structured and post‐selection hypotheses, and multi‐armed bandit problems. Moreover, the BH procedure is a special case of the e‐BH procedure through calibration between p‐values and e‐values. Overall, the e‐BH procedure is a novel, powerful and general tool for multiple testing under dependence, that is complementary to the BH procedure, each being an appropriate choice in different applications.