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Generalized mean p-value merging

Mathematical Methods AIMech:GeneralizedMeanPValueMerging DRAFT

Aliases: Power Mean P-Value Combination, Kolmogorov Generalized Mean P-Merging Function, Hardy-Littlewood-Polya r-Mean P-Value Merging, Vovk-Wang Generalized Mean P-Value Combination

A statistical p-value combination framework (Vovk & Wang 2020) that aggregates K arbitrarily dependent hypothesis test p-values p_1, ..., p_K in [0, 1] using calibrated Hardy-Littlewood-Polya (Kolmogorov) generalized r-means M_{r,K}(p_1, ..., p_K) = (K^(-1) sum_{k=1}^K p_k^r)^(1/r) (and simplex-weighted generalizations M_{r,w}(p_1, ..., p_K) = (sum_{k=1}^K w_k p_k^r)^(1/r) for r in [-infinity, +infinity]) scaled by optimal worst-case multipliers a_{r,K} derived via dual robust risk-aggregation and joint mixability theory so that sup_{U_1, ..., U_K in U} P(a_{r,K} M_{r,K}(U_1, ..., U_K) <= epsilon) <= epsilon holds non-asymptotically for all epsilon in (0, 1) without distributional dependence assumptions.

Identity Notes & Concept Scope No normalized identity match for Generalized Mean P-Value Merging across all 442 pinned AIO records and 568 maintained IDs, names, aliases and scoped synonyms. Fixed-string searches across data, sources, research, history, and reviews returned only frontier references in predecessor discovery reports and a bibliography mention in AIMech:ExactHarmonicMeanPValueTest. Distinct from AIMech:ExactHarmonicMeanPValueTest (Wilson 2019; Fang et al. 2023), which calibrates the unscaled harmonic mean (r = -1) using the Landau stable law S(1, 1, pi/2, mu_n) under independence or regularly varying bivariate Gaussian tail asymptotics rather than Vovk & Wang's exact finite-K worst-case dependence multiplier a_K^H = (y_K + K)^2 / ((y_K + 1) K) <= e * ln K across all joint copulas. Distinct from AIMech:LogarithmicOpinionPool, which aggregates probability distributions via geometric means rather than calibrating hypothesis test p-values under arbitrary dependence via multiplier e (r = 0).

Claims & Grounded Evidence

4 Claims

Mechanism #integrable-exponent-merging-and-finite-sample-precision-interval

For integrable exponents r in (-1, +infinity], scaling the generalized r-mean M_{r,K} by (r + 1)^(1/r) (with continuous limits e * M_{0,K} for the geometric mean at r = 0, 2 * M_{1,K} for the arithmetic mean at r = 1, and M_{infinity,K} = max_k p_k at r = +infinity) yields a valid and asymptotically precise family of p-merging functions under arbitrary dependence (Proposition 3); the multiplier (r + 1)^(1/r) is finite-sample precise (attaining sup P(F <= epsilon) = epsilon for all epsilon in (0, 1)) if and only if r in [1/(K - 1), K - 1] (Proposition 4), whereas for all r in [1, +infinity) and K >= 2 the sharpened multiplier min(r + 1, K)^(1/r) M_{r,K} is finite-sample precise (Proposition 5).

supports Source: Combining p-values via averaging · Locator: Section 3.2, lines 76–111, Theorem 2 and Propositions 3–5; Appendix A, lines 860–896

Mechanism #non-integrable-negative-exponent-and-harmonic-worst-case-multipliers

For heavy-tailed non-integrable exponents r in [-infinity, -1), scaling M_{r,K} by (r / (r + 1)) K^(1 + 1/r) (recovering the Bonferroni correction K * min_k p_k at r = -infinity) is valid and asymptotically precise under arbitrary dependence (Proposition 8; Lemma 13 and Proposition 14); minimizing (r / (r + 1)) K^(1 + 1/r) over r < -1 yields the optimal exponent r* = ln K / (1 - ln K) < -1 and the conservative harmonic bound (e * ln K) M_{-1,K}, while for every K > 2 the exact finite-K precise harmonic multiplier is a_K^H = (y_K + K)^2 / ((y_K + 1) K) <= e * ln K (with a_K^H / ln K -> 1 as K -> infinity), where y_K > 0 is the unique positive root of y(y + K) / ((y + 1) K) = ln(y + 1) (Proposition 9; Appendix A, lines 939–955).

supports Source: Combining p-values via averaging · Locator: Section 3.2, lines 120–149, Theorem 7 and Propositions 8–10; Appendix A, lines 907–981
supports Source: Heavy-tailed distribution for combining dependent p-values with asymptotic robustness · Locator: Section 1, lines 17–21; Section 3.2, lines 86–92

Mechanism #simplex-weighted-generalized-mean-precision-conditions

For prior weights w = (w_1, ..., w_K) in the probability simplex Delta_K with maximum weight w_max = max_k w_k and weighted r-mean M_{r,w}(p_1, ..., p_K) = (sum_{k=1}^K w_k p_k^r)^(1/r), the merging function (r + 1)^(1/r) M_{r,w} is valid for all r > -1 and is finite-sample precise if and only if w_max <= 1/2 and r in [w_max / (1 - w_max), (1 - w_max) / w_max] (Proposition 4w); for r in [1, +infinity), min(r + 1, 1 / w_max)^(1/r) M_{r,w} is finite-sample precise for every weight vector w in Delta_K, reducing when r = 1 and w_1 = w_max > 1/2 to p_1 + sum_{k=2}^K (w_k / w_1) p_k (Proposition 5w).

supports Source: Combining p-values via averaging · Locator: Section 4, lines 151–172, Theorem 2w and Propositions 4w–5w; Appendix A, lines 860–896

Application #power-index-phase-transition-and-compound-bonferroni-averaging

Under exchangeable alternative p-values with power index Pi = sup{m >= 0 : E[p_1^(-m)] < infinity} (where Pi = 1 / (1 - rho^2) > 1 for equicorrelated Gaussian z-tests with correlation rho in (0, 1), Lemma 15), the unscaled r-mean P_{r,K} converges in probability to 0 if r < -Pi, to a non-zero finite limit if r >= -Pi, and the Bonferroni-scaled r-mean K^(1+1/r) P_{r,K} diverges to +infinity if r > -Pi (Proposition 11); because selecting r adaptively from data invalidates Type I error, the compound Bonferroni-Arithmetic F_K^BA = 2 min(K min_k p_k, 2 M_{1,K}) and Bonferroni-Geometric F_K^BG = 2 min(K min_k p_k, e M_{0,K}) merging functions provide asymptotically precise validity across both weak and strong correlation regimes (Proposition 12).

supports Source: Combining p-values via averaging · Locator: Section 5, lines 173–220, Proposition 11; Section 6, lines 221–309, Proposition 12 and Equations 17–18; Appendix A, Lemma 15, lines 983–1024
supports Source: Aggregating Dependent Signals with Heavy-Tailed Combination Tests · Locator: Section 1, lines 50–78

⚠️ Curated Limitations & Negative Results

Because generalized mean p-merging functions guarantee Type I error control over the worst-case dependence structure across all joint copulas of (U_1, ..., U_K), their penalty multipliers (such as 2 for arithmetic mean, e for geometric mean, and a_K^H ~ ln K for harmonic mean) are substantially more conservative than independence-based (Fisher, Stouffer) or bivariate-normal asymptotically robust (Cauchy, Landau harmonic mean) calibrations when test statistics are weakly dependent (Fang et al. 2023, Section 3.2 Remark).
For finite K >= 2 and integrable exponents r outside [1/(K - 1), K - 1] (or simplex weights with max_k w_k > 1/2), the asymptotic multiplier (r + 1)^(1/r) is strictly conservative rather than finite-sample precise (Propositions 4 and 4w), and for r in (-1, 1/(K - 1)) the exact finite-K multiplier requires solving a non-closed-form dual optimization problem (Proposition 6).
Selecting the exponent r post hoc after inspecting the observed p-values invalidates Type I error control; combining multiple exponents requires an explicit multiplicity penalty such as the factor of 2 in the compound Bonferroni-Arithmetic or Bonferroni-Geometric merging functions (Section 6, Proposition 12).
All 1,046 lines of Vovk & Wang (2020, arXiv:1212.4966) were read. In the ar5iv HTML-to-markdown conversion, display-mode equations in Proposition 6 were omitted by the converter while all inline formulas in Propositions 3–5, 8–10, 4w–5w, 11–12, and Appendix A proofs were preserved verbatim. Constructed numerical checks in research/discovery/generalized-mean-pvalue-merging-boundaries.py verify the Proposition 4/5/5w precision boundaries, Proposition 8/9 exact harmonic multipliers, and Section 6 Equation 17 compound merging behavior.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:DataDredgingBias` Data Dredging Bias	related_match	Generalized mean p-value merging combines K arbitrarily dependent hypothesis test p-values with exact non-asymptotic Type I error control across all copulas, preventing false-positive inflation when screening or averaging multiple correlated tests (AIO:DataDredgingBias). AIO:DataDredgingBias denotes the multiple-testing bias rather than the p-value merging method.

Evidence Sources & Primary Literature

3 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`vovk-wang2020`	Combining p-values via averaging Vladimir Vovk, Ruodu Wang	primary paper full_text	ID: https://doi.org/10.1093/biomet/asaa027 sha256:795de89f2d3a…
`fang-chang-park-tseng2023`	Heavy-tailed distribution for combining dependent p-values with asymptotic robustness Yusi Fang, Chung Chang, Yongseok Park, George C. Tseng	primary paper full_text	ID: https://doi.org/10.5705/ss.202021.0078 sha256:432b18ec119b…
`gui-jiang-wang2023`	Aggregating Dependent Signals with Heavy-Tailed Combination Tests Lin Gui, Yuchao Jiang, Jingshu Wang	primary paper full_text	ID: https://doi.org/10.48550/arXiv.2310.20460 sha256:3f487fc6bca4…

Ontology & Grounding Topology

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Curation Provenance

Curator: Antigravity

Method: ai_assisted

Model: Gemini 3

Created: 2026-10-17T10:35:00Z

Updated: 2026-10-17T10:35:00Z

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

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