More powerful multiple testing under dependence via randomization

We show that two procedures for false discovery rate (FDR) control -- the Benjamini-Yekutieli (BY) procedure for dependent p-values, and the e-Benjamini-Hochberg (e-BH) procedure for dependent e-values -- can both be improved by a simple randomization involving one independent uniform random variable. As a corollary, the Simes test under arbitrary dependence is also improved. Importantly, our randomized improvements are never worse than the originals, and typically strictly more powerful, with marked improvements in simulations. The same techniques also improve essentially every other multiple testing procedure based on e-values.