Bayes optimal p-value weighting under a Gaussian prior on the effect sizes
Mathematical Methods AIMech:BayesOptimalPValueWeightingUnderGaussianPrior DRAFT
Aliases: Bayes P-Value Weights, Gaussian-Prior Optimal P-Value Weighting, Uncertainty-Regularized Spjotvoll P-Value Weighting, Informed Genome-Wide Association Study Bayes Weighting (iGWAS)
A frequentist multiple hypothesis testing p-value weighting methodology (and informed genome-wide association study procedure, iGWAS) introduced by Dobriban, Fortney, Kim, & Owen (2015, Biometrika 102(4):753-766, arXiv:1504.02935v1, DOI:10.1093/biomet/asv050) for testing J one-sided Gaussian mean hypotheses H_i: mu_i >= 0 against mu_i < 0 using test statistics T_i | mu_i ~ N(mu_i, 1) and one-sided p-values P_i = Phi(T_i) at per-test significance level q in [0, 1] (controlling the per-family and family-wise error rates strongly at alpha = J q, or false discovery rate via weighted Benjamini-Hochberg) when prior effect-size information is uncertain and modeled by independent Gaussian priors mu_i ~ N(eta_i, sigma_i^2). Whereas classical Spjotvoll p-value weighting (Spjotvoll 1972; Wasserman & Roeder 2006; Roeder & Wasserman 2009; Rubin et al. 2006) assumes exact known effect sizes (sigma_i = 0) and maximizes a strictly concave objective, positive prior variance sigma_i > 0 induces marginal test statistics T_i ~ N(eta_i, gamma_i^2) with gamma_i = sqrt(sigma_i^2 + 1) > 1, rendering the marginal expected discovery objective sum_{i=1}^J Phi((Phi^{-1}(q w_i) - eta_i) / gamma_i) over w_i in [0, 1/q], sum_{i=1}^J w_i = J non-concave (each summand has a concave interval followed by a convex interval). Despite this non-concavity, whenever q satisfies the mild small-q condition sum_{i=1}^J Phi(c_1(eta_i, gamma_i; 1)) >= J q (Condition 5, Proposition 3.4), the exact global Bayes optimal weights take the closed form w_i = Phi(c_1(eta_i, gamma_i; lambda)) / q where c_1(eta_i, gamma_i; lambda) = (-eta_i - gamma_i sqrt(eta_i^2 + 2(gamma_i^2 - 1) log(gamma_i lambda))) / (gamma_i^2 - 1) and lambda >= 1 is the unique dual multiplier found in O(J) cost per Newton step (Theorem 3.2), recovering the Spjotvoll weights Phi(eta_i / 2 + log(lambda) / eta_i) / q as sigma_i -> 0+ (Proposition 3.3); for general q in [0, 1] when lambda in (0, 1), sorting the unique jump-discontinuity thresholds k_i = k(eta_i, gamma_i) in [l(eta_i, gamma_i), 1] where each summand's global maximum jumps from c_1(eta_i, gamma_i; k_i) to +infinity (weight 1/q) solves the non-convex problem in O(J log J) steps for a nearby level q* with |q* - q| <= 1/(2J) (Theorem 3.5).
Identity Notes & Concept Scope
No normalized identity match for Bayes optimal p-value weighting under a Gaussian prior on the effect sizes across all 442 pinned AIO records and 590 maintained IDs, names, aliases and scoped synonyms. Only frontier reference matches in Round 194 and Round 195 discovery artifacts were found. Distinct from AIMech:OptimalMultiWeightedStepUpProcedureForFDRControl (Roquain & van de Wiel 2009), which assumes known alternative p-value CDFs F_i with strictly decreasing densities (strictly concave F_i) to construct a rejection-proportion-indexed multi-weight function W^*(u) for step-up and step-down FDR control, whereas Dobriban, Fortney, Kim, & Owen (2015) model uncertain prior effect sizes mu_i ~ N(eta_i, sigma_i^2) with gamma_i = sqrt(sigma_i^2 + 1) > 1, proving that prior uncertainty makes the static p-value weight optimization non-convex and deriving the exact quadratic-root critical values c_1(eta_i, gamma_i; lambda) for lambda >= 1 (Theorem 3.2) and the O(J log J) jump-discontinuity solver over k_i in [l(eta_i, gamma_i), 1] for lambda < 1 (Theorem 3.5); and distinct from AIMech:AdaptiveDataDrivenOptimalPValueWeighting (Durand 2019, ADDOW / crADDOW) and AIMech:CovariatePoweredCrossWeightedMultipleTesting (Ignatiadis & Huber 2021, IHW), which estimate data-driven group or covariate weights directly from observed p-values.
Claims & Grounded Evidence
4 Claims
Mechanism
#sparse-means-two-regime-phase-transition-and-power-hotspot
In the two-class sparse Gaussian means model where fraction pi_1 of J hypotheses have known negative effect M < 0 and fraction pi_0 = 1 - pi_1 have zero effect (Section 3.2, Proposition 3.1, Section 9.1), optimal p-value weights maximizing expected power at level q exhibit a phase transition at pi_1 Phi(-|M|/2) = q: when |M| is small enough that pi_1 Phi(-|M|/2) > q, all weight is placed on the large effects (w_1 = 1/pi_1, w_0 = 0) with optimal power p^* = pi_1 Phi(Phi^{-1}(q/pi_1) + |M|), whereas when |M| is large enough that pi_1 Phi(-|M|/2) <= q, the interior stationary point c_1 = -|M|/2 of Phi(c - M) - Phi(c) is feasible so w_1 = Phi(-|M|/2)/q < 1/pi_1 and strictly positive weight w_0 = (q - pi_1 Phi(-|M|/2)) / (pi_0 q) > 0 is allocated to zero-mean hypotheses because large effects are already detected with high probability, yielding a 3-to-4-fold power boost over unweighted Bonferroni in the sparse moderate-effect hotspot (Figure 1a).
Mechanism
#exact-small-q-bayes-weights-and-spjotvoll-convergence
Under uncertain Gaussian prior effect sizes mu_i ~ N(eta_i, sigma_i^2) with marginal test standard deviation gamma_i = sqrt(sigma_i^2 + 1) > 1, each summand w_i -> Phi((Phi^{-1}(q w_i) - eta_i) / gamma_i) of the marginal expected discovery objective is non-concave (strictly concave followed by strictly convex), yet for any dual multiplier lambda >= 1 the generic Lagrangian term f(c; eta_i, gamma_i) = Phi((c - eta_i) / gamma_i) - lambda Phi(c) attains its unique global maximum on R union {+infinity} at the smaller quadratic root c_1(eta_i, gamma_i; lambda) = (-eta_i - gamma_i sqrt(eta_i^2 + 2(gamma_i^2 - 1) log(gamma_i lambda))) / (gamma_i^2 - 1) (Lemma 9.1); consequently, whenever q satisfies the mild small-q feasibility condition sum_{i=1}^J Phi(c_1(eta_i, gamma_i; 1)) >= J q (Condition 5, Proposition 3.4), the unique root lambda >= 1 solving sum_{i=1}^J Phi(c_1(eta_i, gamma_i; lambda)) = J q yields the exact global Bayes optimal weights w_i = Phi(c_1(eta_i, gamma_i; lambda)) / q in O(J) cost per Newton iteration (Theorem 3.2), and for any eta_i < 0 the Bayes weight converges smoothly to the deterministic Spjotvoll weight Phi(eta_i / 2 + log(lambda) / eta_i) / q as sigma_i -> 0+ (Proposition 3.3, Section 9.3).
Mechanism
#infinite-variance-symmetry-breaking-and-general-q-jump-solver
As all prior standard errors sigma_i -> infinity (gamma_i -> infinity), non-concavity causes a symmetry-breaking phenomenon where assigning maximum weight 1/q to floor(J q) hypotheses (for non-integer J q) and distributing the remaining budget equally achieves limiting expected power (J + floor(J q)) / 2, strictly exceeding the limiting power J / 2 of uniform weighting w_i = 1 (Section 3.3); for general q in [0, 1] when Condition (5) fails (lambda in (0, 1)), each Lagrangian summand switches its global maximum from c_1(eta_i, gamma_i; lambda) to +infinity (weight 1/q) at a unique threshold k_i = k(eta_i, gamma_i) in [l(eta_i, gamma_i), 1] where l(eta_i, gamma_i) = gamma_i^{-1} exp(-eta_i^2 / (2(gamma_i^2 - 1))) and d(k_i; eta_i, gamma_i) = f(c_1(eta_i, gamma_i; k_i)) - (1 - k_i) = 0 (Lemma 9.2), enabling Algorithm 2 to sort the thresholds k_i and binary search in O(J log J) operations to produce weights that are globally optimal for a nearby level q* satisfying |q* - q| <= 1/(2J) (Theorem 3.5, Section 9.5).
Mechanism
#worst-case-maximin-advantage-and-frequentist-type-i-error-validity
Accounting for prior uncertainty sigma_i > 0 regularizes p-value weights by shrinking extreme Spjotvoll weights nonlinearly toward 1, preventing the severe power loss suffered by deterministic Spjotvoll weighting when it places zero weight on small prior means under uncertain effect sizes and giving Bayes weights a worst-case maximin power/regret advantage across sigma in {0, 1} as well as higher reliability than exponential weighting or two-stage top-candidate filtering (Section 4.1–4.2, Figures 3–4); furthermore, because the Bayes weights w_i >= 0 depend only on independent prior data (such as a prior GWAS with eta_i = (N_i / N_{0i})^{1/2} T_{0i} and sigma_i^2 = phi N_i / N_{0i} in iGWAS) and satisfy sum_{i=1}^J w_i <= J, weighted Bonferroni (or weighted Benjamini-Hochberg) retains strong frequentist PFER and FWER control at alpha = J q without requiring independence of the current test statistics T_i and even if the Gaussian prior or pleiotropy model is completely misspecified (Section 3.1, Section 5.2–5.3).
⚠️ Curated Limitations & Negative Results
- When the significance level q is large enough that Condition (5) fails (lambda* < 1), the Lagrangian sum of weights W(lambda) has jump discontinuities of size (1 - r_i)/q at the thresholds k_i = k(eta_i, gamma_i), so an exact dual solution satisfying W(lambda) = J may not exist; in Case 2 of Section 9.5.2, Algorithm 2 instead returns weights that are globally optimal for a perturbed level q* within |q* - q| <= 1/(2J) (Theorem 3.5).
- Valid frequentist type I error control requires that the prior summary statistics (eta_i, sigma_i) be statistically independent of the current study test statistics T_i (for example, shared control cohorts across GWAS studies can violate independence, Section 6.2 & Section 10.5), and in pleiotropic GWAS applications there is no principled a priori rule for tuning the overdispersion parameter phi without external validation data (Section 5.2 & Section 6.5).
- The analytical Bayes weights formula relies on the Gaussian means model T_i | mu_i ~ N(mu_i, 1) with Gaussian prior mu_i ~ N(eta_i, sigma_i^2); non-Gaussian likelihoods or order/box constraints on the weights require general convex-programming approximations such as Princessp (Dobriban 2017).
- All 762 lines of Dobriban, Fortney, Kim, & Owen (2015, Biometrika 102(4):753-766, arXiv:1504.02935v1, DOI:10.1093/biomet/asv050) were read. Constructed numerical checks in research/discovery/bayes-optimal-p-value-weighting-under-gaussian-prior-boundaries.py verify (1) Proposition 3.1 sparse-means two-regime phase transition (Regime 2 positive zero-mean weight w_0 = 1.001538, p^* = 0.002999 > all-on-M power 0.002, and hotspot power ratio = 3.052 > 3.0), (2) Theorem 3.2 exact small-q Bayes weights (lambda^* = 6.726987, budget error = 5.329070518200751e-15, global Lagrangian optimality = True) and Proposition 3.3 convergence to Spjotvoll weights as sigma -> 0+ (max diff shrinking from 0.695122152 at sigma = 1.0 to 1.98e-07 at sigma = 1e-4), (3) Section 3.3 sigma -> infinity symmetry breaking (power = 5.99983 vs. uniform 5.00013) and Theorem 3.5 / Lemma 9.2 general-q jump solver (max root error = 1.1102230246251565e-16, |q^* - q| = 0.025041 <= 1/(2J) = 0.0625), and (4) Section 4.2 maximin advantage over Spjotvoll weights (Bayes worst-case regret = 0.009176 < Spjotvoll regret = 0.026571, min relative efficiency = 0.857032 > 0.686113) and exact frequentist FWER control under prior misspecification (FWER = 0.0456 <= 0.05).
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:DataDredgingBias
Data Dredging Bias
|
related_match | Bayes optimal p-value weighting allocates per-hypothesis significance thresholds q w_i using independent Gaussian prior effect-size distributions while strongly controlling the frequentist family-wise error rate or false discovery rate to prevent false-positive multiplicity inflation (AIO:DataDredgingBias). |
Evidence Sources & Primary Literature
3 Sources| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
dobriban2015-bayesweights-ar5iv
|
Edgar Dobriban, Kristen Fortney, Stuart K. Kim, Art B. Owen
|
primary paper
full_text
|
ID: https://doi.org/10.1093/biomet/asv050
sha256:74f78d99ae84…
|
dobriban2015-bayesweights-crossref
|
Edgar Dobriban, Kristen Fortney, Stuart K. Kim, Art B. Owen
|
primary paper
abstract_only
|
ID: https://doi.org/10.1093/biomet/asv050
sha256:bb77cf0915e8…
|
dobriban2015-bayesweights-abs
|
Edgar Dobriban, Kristen Fortney, Stuart K. Kim, Art B. Owen
|
primary paper
abstract_only
|
ID: https://doi.org/10.48550/arXiv.1504.02935
sha256:06609ec82532…
|
Ontology & Grounding Topology
Open in interactive network graph →Curation Provenance
Curator: Antigravity
Method: ai_assisted
Model: Gemini 3
Created: 2026-10-17T15:10:00Z
Updated: 2026-10-17T15:10:00Z
Source File:
data/curated/BayesOptimalPValueWeightingUnderGaussianPrior.yaml
· Schema:
curated_concept.yaml
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