Optimal Certified Multiple Testing with E-values
Abstract
We study randomized multiple testing with e-values under arbitrary dependence and introduce a conditional certification constraint: Given the observed evidence, each coordinate's expected rejection certificate must not exceed its e-value. Existing randomized e-BH improvements satisfy this constraint but have no established optimality guarantee. We prove that the maximum conditional inclusion probability for hypothesis i, given e-values~x, is 1 ∧ (α/ K) ∑j min(xi, xj), and construct one rule attaining these bounds simultaneously. The rule uses a single independent uniform variable, rejects a prefix of the ordered e-values, and has a unique conditional rejection-set law among rules attaining all the bounds. It therefore maximizes every nonnegative weighted sum of inclusion probabilities, including expected true discoveries under any truth configuration. Under a shared uniform draw, its rejection set contains those of e-BH and U-eBH. The construction extends to BY-calibrated p-values, and simulations under positive and negative dependence show power gains over existing randomized procedures.
Related
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BibTeX
@misc{bose2026optimal,
title = {{Optimal Certified Multiple Testing with E-values}},
author = {Soumyabrata Bose and Jay Bartroff},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01333},
eprint = {2611.01333},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01333},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal Certified Multiple Testing with E-values
AU - Soumyabrata Bose
AU - Jay Bartroff
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01333
UR - https://arxiv.org/abs/2611.01333
AB - We study randomized multiple testing with e-values under arbitrary dependence and introduce a conditional certification constraint: Given the observed evidence, each coordinate's expected rejection certificate must not exceed its e-value. Existing randomized e-BH improvements satisfy this constraint but have no established optimality guarantee. We prove that the maximum conditional inclusion probability for hypothesis i, given e-values~x, is 1 ∧ (α/ K) ∑_j min(x_i, x_j), and construct one rule attaining these bounds simultaneously. The rule uses a single independent uniform variable, rejects a prefix of the ordered e-values, and has a unique conditional rejection-set law among rules attaining all the bounds. It therefore maximizes every nonnegative weighted sum of inclusion probabilities, including expected true discoveries under any truth configuration. Under a shared uniform draw, its rejection set contains those of e-BH and U-eBH. The construction extends to BY-calibrated p-values, and simulations under positive and negative dependence show power gains over existing randomized procedures.
ER -