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E-value calibration and merging

Mathematical Methods AIMech:EValueCalibrationAndMerging DRAFT

Aliases: E-Value Calibration and Combination Framework, Vovk-Wang E-Merging and P-to-E Calibration, Admissible Symmetric E-Merging and Sequential U-Statistic E-Merging, Closed-Testing Family-Wise Valid E-Value Adjustment

A statistical hypothesis testing and evidence synthesis framework (Vovk & Wang 2021) based on e-variables—nonnegative random variables E >= 0 satisfying E_Q[E] <= 1 under the null hypothesis Q—that characterizes admissible p-to-e calibrators f: [0, 1] -> [0, infinity] via upper semicontinuity, f(0) = infinity, and integral_0^1 f(p) dp = 1 (Proposition 2.1), establishes g(e) = min(1, 1/e) as the unique admissible e-to-p calibrator (Proposition 2.2), proves that under arbitrary dependence the arithmetic mean M_K(e_1, ..., e_K) = (1/K) sum_{k=1}^K e_k essentially dominates all symmetric e-merging functions while {lambda + (1 - lambda) M_K : lambda in [0, 1]} forms the minimal complete class of admissible symmetric e-merging functions (Proposition 3.1, Theorem 3.2), characterizes admissible U-statistic and product e-merging functions under independent or sequential test-supermartingale e-variables (Propositions 4.1–4.2), and derives polynomial-time closed-testing adjustments for family-wise valid multiple testing (Theorem 5.1, Algorithms 1–2).

Identity Notes & Concept Scope No normalized identity match for E-Value Calibration and Merging across all 442 pinned AIO records and 570 maintained IDs, names, aliases and scoped synonyms. Fixed-string searches across data, sources, research, history, and reviews for EValueCalibrationAndMerging, e-merging, p-to-e calibrator, e-to-p calibrator, ie-merging, and p-to-e merging returned only 29 lines matching the substring 'e-merging' inside 'pvalue-merging' from Round 174. Distinct from AIMech:GeneralizedMeanPValueMerging (Vovk & Wang 2020), which combines p-values via Kolmogorov-Nagumo r-means calibrated by penalty multipliers a_{r,K} rather than combining nonnegative e-variables E_Q[E] <= 1 via parameter-free arithmetic averaging or sequential supermartingale products.

Claims & Grounded Evidence

4 Claims

Mechanism #admissible-p-to-e-and-e-to-p-calibrator-characterization

A decreasing function f: [0, 1] -> [0, infinity] is a p-to-e calibrator (mapping every p-variable P to an e-variable f(P) with E_Q[f(P)] <= 1) if and only if integral_0^1 f(p) dp <= 1, and is admissible if and only if f is upper semicontinuous (left-continuous), f(0) = infinity, and integral_0^1 f(p) dp = 1, a class containing the Shafer power calibrators f_kappa(p) = kappa p^(kappa - 1) for kappa in (0, 1) (Eq. 1) and the parameter-free mixture calibrator F(p) = (1 - p + p ln p) / (p (-ln p)^2) (Eq. 2), whereas the Vovk-Sellke bound VS(p) = (-e p ln p)^(-1) for p <= e^(-1) is not a valid calibrator (Proposition 2.1, Remark 2.3); conversely, g(t) = min(1, 1/t) is an e-to-p calibrator by Markov's inequality and strictly dominates every other e-to-p calibrator, making it the unique admissible e-to-p calibrator (Proposition 2.2).

supports Source: E-values: Calibration, combination, and applications · Locator: Section 2, lines 26–74, Propositions 2.1–2.2, Equations 1–2, Remark 2.3; Appendix B, lines 993–1003, Equations B.1–B.2

Mechanism #arithmetic-mean-essential-domination-and-minimal-complete-symmetric-class

Under arbitrary dependence among K e-variables, the arithmetic mean M_K(e_1, ..., e_K) = (1/K) sum_{k=1}^K e_k essentially dominates every symmetric e-merging function F (meaning M_K(e) >= F(e) whenever F(e) > 1) and dominates every symmetric positively homogeneous e-merging function (Proposition 3.1); more generally, every symmetric e-merging function is dominated by F_lambda(e) = lambda + (1 - lambda) M_K(e) for some lambda in [0, 1], and a symmetric e-merging function is admissible if and only if it belongs to the minimal complete class {lambda + (1 - lambda) M_K : lambda in [0, 1]} where lambda = F(0) (Theorem 3.2 and Appendix E, Propositions E.1–E.6); for generalized r-means M_{r,K} under arbitrary dependence, min(K^(1/r - 1), 1) M_{r,K} is a precise e-merging function for all r in [-infinity, infinity] but is admissible if and only if r = 1 (Proposition 6.1).

supports Source: E-values: Calibration, combination, and applications · Locator: Section 3, lines 75–103, Proposition 3.1 and Theorem 3.2; Section 6, lines 170–211, Proposition 6.1 and Equations 18–19; Appendix E, lines 1110–1185, Propositions E.1–E.6

Mechanism #independent-and-sequential-supermartingale-u-statistic-e-merging

When K e-variables E_1, ..., E_K are independent (ie-merging) or sequential with E_Q[E_k | E_1, ..., E_{k-1}] <= 1 (se-merging, so partial products form a nonnegative test supermartingale), the product M^0_K(e_1, ..., e_K) = e_1 * ... * e_K (Eq. 7), the elementary symmetric U-statistics U_n(e_1, ..., e_K) = C(K, n)^(-1) sum_{1<=k_1<...<k_n<=K} e_{k_1} ... e_{k_n} for n in {0, 1, ..., K} (Eq. 8), and all their convex mixtures sum_{n=0}^K lambda_n U_n are admissible ie-merging and se-merging functions (Proposition 4.1), and the product function e_1 * ... * e_K weakly dominates every ie-merging and se-merging function F in that e_1 * ... * e_K >= F(e_1, ..., e_K) for all (e_1, ..., e_K) in [1, infinity)^K (Proposition 4.2).

supports Source: E-values: Calibration, combination, and applications · Locator: Section 4, lines 104–137, Propositions 4.1–4.2 and Equations 7–8

Application #closed-testing-family-wise-valid-multiple-testing-algorithms

Applying the Marcus-Peritz-Gabriel closure principle to intersection e-merging functions F_I yields family-wise valid (FWV) adjusted e-values e*_k = min_{I subseteq {1,...,K}: k in I} F_I(e_I) satisfying E_Q[max_{k in I_0} E*_k] <= 1 for the true null index set I_0 (Section 5, Theorem 5.1, Equations 9–11; Appendix A.3, lines 955–983); for arbitrary dependence with F_I = M_{|I|}, sorting e_(1) <= ... <= e_(K) and evaluating e*_(k) = min_{i in {0,...,k-1}} (sum_{j=1}^i e_(j) + e_(k)) / (i + 1) computes the exact 2^K - 1 closure minimum in O(K^2) time and strictly dominates naive Bonferroni e-adjustment e_k / K (Algorithm 1, Eq. 13), while for sequential or independent e-values with F_I(e_I) = prod_{i in I} e_i, e*_k = e_k prod_{i != k} min(1, e_i) computes the exact closure minimum in O(K) time without sorting (Algorithm 2, Eq. 14).

supports Source: E-values: Calibration, combination, and applications · Locator: Section 5, lines 138–169, Theorem 5.1, Equations 9–14, and Algorithms 1–2; Appendix A.3, lines 955–983; Appendix G, lines 1234–1268, Propositions G.1–G.2

⚠️ Curated Limitations & Negative Results

Round-trip calibration between p-values and e-values is inherently lossy: calibrating a p-value p to an e-value via f_kappa(p) = kappa p^(kappa - 1) and back via min(1, 1/e) yields p' = min(1, p^(1 - kappa) / kappa) > p, and the Vovk-Sellke upper bound VS(p) = (-e p ln p)^(-1) is not a valid p-to-e calibrator because integral_0^1 VS(p) dp = infinity (Section 2, Propositions 2.1–2.2 and Remark 2.3).
Under arbitrary dependence, no symmetric e-merging function can strictly exceed the arithmetic mean M_K(e) in the rejection region M_K(e) >= 1 (Proposition 3.1 and Theorem 3.2), and all other generalized r-means M_{r,K} for r != 1 require worst-case penalty multipliers min(K^(1/r - 1), 1) that make them strictly inadmissible (Proposition 6.1).
Product e-merging e_1 * ... * e_K and Algorithm 2 O(K) closed testing require the sequential supermartingale condition E_Q[E_k | E_1, ..., E_{k-1}] <= 1 (or independence); if e-variables are arbitrarily dependent, product merging can violate E_Q[F(E)] <= 1 by up to K-th power growth, and if likelihood-ratio e-variables are misspecified when the alternative holds only on part of a sequential stream, product merging can lose accumulated evidence compared with universal test martingales (Section 4, Section 7, and Appendix H, lines 1270–1274).
All 1,288 lines of Vovk & Wang (2021, arXiv:1912.06116) were read. Constructed numerical checks in research/discovery/evalue-calibration-and-merging-boundaries.py verify Propositions 2.1–2.2 calibrator integrals, Proposition 3.1 / Theorem 3.2 / Proposition 6.1 domination over scaled r-means, Propositions 4.1–4.2 sequential filtration tree U-statistic expectations, and Section 5 Algorithms 1 and 2 exact agreement with brute-force 2^K - 1 subset closure.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:DataDredgingBias` Data Dredging Bias	related_match	E-value calibration, symmetric arithmetic-mean e-merging, sequential test-supermartingale product merging, and Algorithms 1–2 closed-testing adjustments preserve valid type I error and family-wise validity under optional continuation, post-hoc study combination, and arbitrary dependence, preventing false-positive inflation from data dredging (AIO:DataDredgingBias). AIO:DataDredgingBias denotes the multiple-testing and optional-stopping bias rather than the e-value methodology.

Evidence Sources & Primary Literature

2 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`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…
`vovk-wang2020-biometrika`	Combining p-values via averaging Vladimir Vovk, Ruodu Wang	primary paper full_text	ID: https://doi.org/10.1093/biomet/asaa027 sha256:795de89f2d3a…

Ontology & Grounding Topology

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Curation Provenance

Curator: Antigravity

Method: ai_assisted

Model: Gemini 3

Created: 2026-10-17T11:05:00Z

Updated: 2026-10-17T11:05:00Z

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

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