Home / Curated / Mathematical Methods / E-value Benjamini-Hochberg procedure

E-value Benjamini-Hochberg procedure

Mathematical Methods AIMech:EValueBenjaminiHochbergProcedure DRAFT

Aliases: e-BH Procedure, Base e-BH and Boosted e-BH Multiple Testing Procedure, Self-Consistent E-Value False Discovery Rate Control, Post-Selection and Structured e-BH Procedure

A step-down multiple hypothesis testing procedure (Wang & Ramdas 2022) for controlling the false discovery rate (FDR) at target level alpha in (0, 1) using nonnegative e-variables E_1, ..., E_K satisfying E[E_k] <= 1 under each true null hypothesis H_k (k in N, |N| = K_0) that (1) optionally multiplies each raw e-value e_k by a marginal-distribution-informed boosting factor b_k >= 1 calibrated via the discrete grid truncation function T(x) = max{K/k : k in {1, ..., K}, K/k <= x} under arbitrary dependence (E[T(alpha b_k E_k)] <= alpha, Eq. 9) or under positive regression dependency on a subset (PRDS, max_{j in {1,...,K}} (K/j) P(alpha b_k E_k >= K/j) <= alpha, Eq. 8), and (2) orders the boosted e-values e'_[1] >= e'_[2] >= ... >= e'_[K] to reject the k_e^* hypotheses with the largest boosted e-values where k_e^* = max{k in {1, ..., K} : k e'_[k] / K >= 1 / alpha} (with max(empty) = 0; equivalently e'_k >= t_alpha = inf{t >= 0 : t R(t) / K >= 1 / alpha}, Proposition 1), guaranteeing finite-sample FDR <= alpha K_0 / K under arbitrary dependence without any logarithmic Benjamini-Yekutieli correction (Theorem 2, Proposition 2).

Identity Notes & Concept Scope No normalized identity match for E-value Benjamini-Hochberg procedure across all 442 pinned AIO records and 571 maintained IDs, names, aliases and scoped synonyms. Fixed-string searches across data, sources, research, history, and reviews returned only 7 lines in Round 176 discovery and review artifacts where arXiv:2009.02824 was queued as a pending frontier lead. Distinct from AIMech:EValueCalibrationAndMerging (Vovk & Wang 2021), which merges e-values for global-null testing and controls family-wise error rate (FWER) via Marcus-Peritz-Gabriel closed testing (Algorithms 1–2) rather than controlling the false discovery rate (FDR) via self-consistent step-down thresholding and grid-truncation boosting.

Claims & Grounded Evidence

4 Claims

Mechanism #base-ebh-stepdown-threshold-equivalence-and-self-consistent-fdr-control

Given nonnegative (boosted or raw) e-values e'_1, ..., e'_K sorted as e'_[1] >= ... >= e'_[K], the base e-BH procedure G(alpha) rejects the k_e^* = max{k in {1, ..., K} : k e'_[k] / K >= 1 / alpha} largest e-values (Eq. 5), which is equivalent to rejecting all H_k with e'_k >= t_alpha where t_alpha = inf{t in [0, infinity) : t R(t) / K >= 1 / alpha} in [1/alpha, K/alpha] and R(t) = max(sum_{k=1}^K 1_{e'_k >= t}, 1) (Eq. 6, Proposition 1); furthermore, any self-consistent e-testing procedure G satisfying e_k >= K / (alpha R_G) for every rejected hypothesis controls FDR at level alpha K_0 / K under arbitrary dependence because E[F_G / R_G] = sum_{k in N} E[1_{k in G} / R_G] <= (alpha / K) sum_{k in N} E[E_k] <= alpha K_0 / K (Theorem 2, Proposition 2), and running e-BH on any arbitrarily data-screened subset S subseteq {1, ..., K} at amended level alpha_tilde = alpha |S| / K or over any combinatorial family S_family of structured rejection sets preserves self-consistency and arbitrary-dependence FDR control (Section 5.1).

supports Source: False discovery rate control with e-values · Locator: Section 4.1, lines 160–179, Equations 5–6, Proposition 1; Section 5, lines 199–225, Theorem 2 and Proposition 2; Appendix B, lines 1078–1084

Mechanism #grid-truncation-boosting-under-arbitrary-dependence-and-sharpness

Defining the grid truncation function T(x) = max{K/k : k in {1, ..., K}, K/k <= x} with max(empty) = 0 (Eq. 7) and applying Lemma 1(iii) E[f(X) 1_{X_k >= f(X)}] <= E[X'_k] to f(E') = t_alpha with range (alpha^(-1) K / K_set), the base e-BH procedure applied to nonnegative random variables E'_1, ..., E'_K under arbitrary dependence satisfies FDR_{G(alpha)} = (1 / K) sum_{k in N} E[t_alpha^(-1) 1_{E'_k >= t_alpha}] <= (K_0 / K) y_alpha where y_alpha = (1 / K_0) sum_{k in N} E[T(alpha E'_k)] (Theorem 3, Eq. 16); choosing boosting factors b_k >= 1 via E[T(alpha b_k E_k)] <= alpha (Eq. 9) or the K-free upper bound y_bar_{alpha,k}(b_k) := alpha b_k E[E_k 1_{alpha b_k E_k >= 1}] <= alpha (Eq. 17) ensures FDR <= alpha K_0 / K, yielding b = (2 / alpha)^(1/2) (approx. 6.324555 at alpha = 0.05) for power-calibrator e-values E_k = (1/2) P_k^(-1/2) (Example 2) and solving b Phi(delta/2 + ln(alpha b)/delta) = 1 (b approx. 1.373374 for delta = 3 and 1.11197 for delta = 4 at alpha = 0.05) for Gaussian likelihood-ratio e-values E_k = exp(delta X_k - delta^2/2) (Example 3), while the unboosted bound alpha K_0 / K is attained with equality when E_k = (K / (K_0 alpha)) 1_A for k in N with P(A) = alpha K_0 / K and E_k = 0 otherwise (Example 4).

supports Source: False discovery rate control with e-values · Locator: Section 4.2, lines 180–198, Equations 7–11; Section 6.1, lines 278–326, Lemma 1; Section 6.2, lines 328–358, Theorem 3 and Examples 2–4; Appendix B, lines 1086–1172

Mechanism #prds-fdr-bound-and-exact-quantile-boosting-factor

When the raw e-values satisfy positive regression dependency on a subset (PRDS) on each null coordinate k in N, Lemma 1(ii) implies FDR_{G(alpha)} <= (K_0 / K) z_alpha where z_alpha = (1 / K_0) sum_{k in N} max_{j in {1,...,K}} (K/j) P(alpha E'_k >= K/j) (Theorem 4, Eq. 21); if E_k is continuously distributed and t -> t P(E_k >= t) is decreasing on [q_{1-alpha}(E_k), infinity) (Condition 23), the largest boosting factor satisfying Eq. (8) is exactly b_k = (alpha q_{1-alpha}(E_k))^(-1) (Proposition 3), which boosts power-calibrator e-values E_k = lambda P_k^(lambda - 1) by b = lambda^(-1) alpha^(-lambda) (reducing FDR by a factor z_{alpha,k}(b) / y_bar_{alpha,k}(b) = lambda, and yielding b = 2 alpha^(-1/2) approx. 8.944272 at lambda = 1/2, alpha = 0.05 in Example 5) and boosts Gaussian likelihood-ratio e-values via z_bar_{alpha,k}(b) = sup_{x >= 1} x (1 - Phi(delta/2 + ln(x/(alpha b))/delta)) = alpha to b approx. 7.88 for delta = 3 and b approx. 10.31–10.32 for delta = 4 at alpha = 0.05 (Example 6).

supports Source: False discovery rate control with e-values · Locator: Section 6.3, lines 359–388, Theorem 4, Proposition 3, Equations 21–24, and Examples 5–6; Appendix B, lines 1173–1185

Application #monotone-transform-optimality-and-bh-by-calibration-unification

Among all e-testing procedures G(phi) generated by a strictly increasing continuous transform phi: [0, infinity] -> [0, infinity] that reject the k^*_{e,phi} = max{k : k phi(e_[k]) / K >= 1} largest e-values and guarantee FDR_{G(phi)} <= alpha across all configurations of e-values, G(phi) is always a subset of the base e-BH procedure G(alpha) (Theorem 5, proved by constructing k_0-simultaneous indicator null e-values E_k = m' 1_{A_k} whenever phi^(-1)(K/k_0) < K/(k_0 alpha)); furthermore, applying e-BH to decreasing p-value transforms psi(P_k) yields step-up p-testing procedures D(psi) with FDR <= (K_0 / K) E[T(psi(P))] under arbitrary dependence and FDR <= (K_0 / K) sup_{t >= 1} t psi^(-1)(t) under PRDS (Proposition 4), recovering both the Benjamini-Hochberg procedure (FDR <= alpha K_0 / K under PRDS) and the Benjamini-Yekutieli procedure (FDR <= alpha l_K K_0 / K under arbitrary dependence, where l_K = sum_{i=1}^K 1/i) via psi(p) = alpha / p.

supports Source: False discovery rate control with e-values · Locator: Section 6.4, lines 389–401, Theorem 5; Section 6.5, lines 402–441, Propositions 4–5 and Example 7; Appendix B, lines 1187–1217; Appendix C, lines 1218–1231, Proposition 6

⚠️ Curated Limitations & Negative Results

When marginal null distributions of the e-values are unknown so that boosting must default to b_k = 1, the actual FDR bound (K_0 / K) E[T(alpha E_k)] of the unboosted base e-BH procedure is typically strictly smaller than alpha K_0 / K because T(alpha E_k) truncates values below 1 to 0, resulting in conservative power compared with distribution-boosted e-BH (Section 5.2 Remark 1 and Section 6.2 Example 2).
When exact uniform p-values are available and satisfy positive regression dependency on a subset (PRDS), the classical Benjamini-Hochberg procedure D(alpha) dominates any p-testing procedure D(psi) obtained by calibrating p-values via a transform psi satisfying Condition 29 (Section 6.5 Proposition 5), and under arbitrary dependence the Benjamini-Yekutieli (BY) procedure can yield more rejections than power-calibrated e-BH when the discovery proportion k^* / K is smaller than approximately 2 / log K (Section 6.5 Example 7 and Appendix A Table 4(b)).
Because heterogeneous boosting factors b_k >= 1 depend on each hypothesis's marginal null distribution, the boosted e-values e'_k = b_k e_k need not preserve the ordering of the raw e-values e_k, so the full e-BH procedure can reject a hypothesis with a smaller raw e-value while failing to reject one with a larger raw e-value (Section 5.2 Item 4).
All 1,242 lines of Wang & Ramdas (2022, arXiv:2009.02824) were read. Constructed numerical checks in research/discovery/evalue-benjamini-hochberg-procedure-boundaries.py verify Proposition 1 step-down/threshold equivalence, Section 5.1 post-selection self-consistency, Example 4 sharpness of the alpha K_0 / K bound, Examples 2, 3, 5, and 6 boosting factors under arbitrary dependence and PRDS, the Theorem 5 optimality counterexample, and Proposition 4 exact recovery of the BY harmonic factor l_K.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:DataDredgingBias` Data Dredging Bias	related_match	The e-BH procedure and its self-consistent post-selection and structured variants control the false discovery rate under arbitrary dependence, optional stopping of bandit e-processes, and arbitrary data-dependent hypothesis screening without false-positive inflation from data dredging (AIO:DataDredgingBias). AIO:DataDredgingBias denotes the multiple-testing and selective-inference bias rather than the e-BH statistical testing procedure.

Evidence Sources & Primary Literature

2 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`wang-ramdas2022-ebh`	False discovery rate control with e-values Ruodu Wang, Aaditya Ramdas	primary paper full_text	ID: https://doi.org/10.1111/rssb.12489 sha256:26bd39717641…
`vovk-wang2021-evalues`	E-values: Calibration, combination, and applications Vladimir Vovk, Ruodu Wang	primary paper full_text	ID: https://doi.org/10.1214/20-AOS2020 sha256:9ca237ebcb82…

Ontology & Grounding Topology

Open in interactive network graph →

Curation Provenance

Curator: Antigravity

Method: ai_assisted

Model: Gemini 3

Created: 2026-10-17T11:20:00Z

Updated: 2026-10-17T11:20:00Z

Source File: data/curated/EValueBenjaminiHochbergProcedure.yaml · Schema: curated_concept.yaml

Explore across X-Mech

Related computational & biological Mechs: Embedding Map Network Graph ProteinTraitsMech (ESM-2 pLM) DUFMech (Domain pLM) TraitMech CultureMech All 14 X-Mech Repositories