Pith. sign in

REVIEW 1 major objections 4 minor 20 references

Weighted-mean eBH procedures are the only admissible FDR controllers

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Weighted-mean closed eBH procedures form the complete admissible class for FDR control with e-values.

T0 review reviewed 2026-08-02 challenge →

load-bearing objection First complete-class theory for FDR e-values; central claim likely right, but one unproved representation lemma is load-bearing. the 1 major comments →

arxiv 2607.14380 v1 pith:SGQFVK2B submitted 2026-07-15 stat.ME

Admissibility and Complete Classes for False Discovery Rate Control with E-values

Liulei Sun , Ruodu Wang This is my paper
classification stat.ME MSC 62C1562J15
keywords e-valuesfalse discovery rateadmissibilitycomplete classe-Benjamini-Hochbergmultiple testingweighted-mean eBHarbitrary dependence
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks which procedures for controlling the false discovery rate with e-values can be improved. It answers by showing that the weighted-mean closed eBH procedures—one-parameter families built from weighted averages of input e-values—are the only procedures that matter: every valid simultaneous procedure is strongly dominated by an admissible member of this class, and the same holds for point procedures that output a single rejection set. If true, this means no FDR-controlling e-value procedure outside this family can be admissible, and the design of powerful procedures reduces to choosing weights and constant terms inside the family. The paper also draws sharp boundaries: constant-free weighted-mean eBH procedures are admissible at every level, while the point version is admissible only below level 1/2.

Core claim

The central discovery is a complete class theorem: at any FDR level alpha, for any simultaneous procedure D that controls FDR under arbitrary dependence, there exists an admissible weighted-mean closed eBH procedure that strongly dominates D. Consequently every admissible simultaneous procedure is a weighted-mean eBH procedure; point procedures have the analogous statement with point weighted-mean eBH procedures. The companion admissibility theorems show the family is not merely complete but filled with acceptable members: every constant-free weighted-mean eBH procedure is admissible at every level, and the point counterpart is admissible for alpha<1/2 with this threshold sharp for the mean

What carries the argument

The load-bearing object is the weighted-mean e-merging function M_lambda(e)=lambda_0+sum_{i in A} lambda_i e_i for a subset A, with weights in the simplex and infinity handled by convention. Substituting these functions for the e-collection in the closed eBH procedure yields the weighted-mean eBH procedure. The admissibility proofs work by constructing a violated FDR bound through a mixture of e-value vectors, and the complete-class proof rests on a representation lemma stating that any bounded function F with E[F(X)]<=1 for all finitely supported X with E[X_i]<=1 is dominated by a weighted-mean function.

Load-bearing premise

The complete-class proof depends on an unproved representation lemma: any bounded function F on the positive orthant with E[F(X)]<=1 for every finitely supported X with E[X_i]<=1 is dominated by a weighted-mean function; if that lemma fails, the domination step collapses.

What would settle it

Find a bounded function F on [0,infinity)^m such that E[F(X)]<=1 for every finitely supported X with E[X_i]<=1 but F is not bounded above by any function of the form lambda_0 + sum lambda_i x_i; such a counterexample would invalidate the domination step in the complete-class theorem.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Constant-free asymmetric weights can encode prior information about hypotheses without sacrificing admissibility.
  • The point mean eBH procedure is admissible exactly for alpha<1/2; above that threshold a uniform improvement exists.
  • Within the symmetric simultaneous class, mean eBH is the largest element exactly when alpha<1/K; otherwise the class has no largest element.
  • The closed BY procedure is inadmissible, so closure alone does not guarantee admissibility—the choice of e-collection is essential.
  • For symmetric weighted-mean eBH with constant terms, a sharp admissibility boundary is identified, giving explicit guidance for choosing constants.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the representation lemma holds more broadly, the complete class theorem would extend to procedures defined on the full space of e-collections rather than only vectors of e-values.
  • The result suggests that in practice, learning weights from data is not only a power boost but can be done without compromising the decision-theoretic guarantee of being unimprovable, as the simulations illustrate.
  • The sharp alpha<1/2 threshold for point mean eBH echoes a general phenomenon in simultaneous versus point decision problems, and similar thresholds may appear in other FDR settings.
  • For p-value-based FDR procedures, the paper's framework does not directly apply; an analogous complete class theory would need different tools, possibly via p-to-e calibration.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper develops a decision-theoretic framework for FDR control with e-values under arbitrary dependence, distinguishing simultaneous procedures (outputting collections of rejection sets) from point procedures. It introduces weighted-mean closed eBH procedures and proves: (i) constant-free weighted-mean eBH procedures are admissible at every level (Theorem 4.1); (ii) point analogues are admissible for alpha<1/2 under strict positivity (Theorem 4.2); (iii) every simultaneous FDR-controlling procedure is strongly dominated by an admissible weighted-mean eBH procedure, so these form a complete class (Theorem 5.1), with an analogous point result (Theorem 5.3); (iv) within the symmetric class the mean eBH procedure is the largest exactly for alpha<1/K, and constant-term conditions for admissibility are derived (Theorems 6.3, 6.5, 6.6). The closed BY procedure is shown to be inadmissible, and a simulation studies learned asymmetric weights.

Significance. If the main theorem is valid, this is the first complete-class characterization for FDR control in e-value testing. The paper gives explicit admissible families, sharp thresholds, and counterexamples (B.1--B.4) that clarify the definitions. The admissibility proofs are largely self-contained, and the simulation includes a useful negative control. However, the central complete-class result depends on an unproved representation lemma, so the contribution is currently conditional.

major comments (1)
  1. [Theorem 5.1, Step 1] The complete-class proof depends on the displayed 'useful fact' in Section 5.1: every bounded F on [0,∞)^m with E[F(X)]≤1 for all finitely supported X with E[X_i]≤1 is dominated by some M_lambda. The paper says 'the same proof idea applies after a simple adjustment, and we omit the details,' citing Clerico (2026). This is not a routine rewording: F_A(e)=sup_{R in D(e)} |A∩R|/(alpha(|R|∨1)) is only known bounded and is built from an arbitrary measurable D, so the lemma must hold without monotonicity or continuity assumptions. The use of the lemma is exactly what converts pointwise FDR control of D into membership in a weighted-mean eBH procedure (the inequality f_A(e) ≤ E^lambda_A(e)); without it, Steps 2--3, Theorem 5.3, Proposition 2.7, and Propositions 6.1--6.2 do not go through. Please supply a complete proof of the useful fact, or a precise theorem statement from Clerico (2026) with
minor comments (4)
  1. [Section 7] The statement 'Theorem 4.1 therefore implies that the resulting weighted-mean eBH procedure is admissible' is not formally covered by the framework, since the weights are learned from training data and the procedure is therefore a data-dependent mapping rather than a fixed element of SP_alpha. Please clarify that admissibility is meant conditionally on the realized weights, or extend the formalism to randomized/adaptive procedures.
  2. [Theorem 4.2 proof] The symbol q is used for different quantities in Case 1 and Case 2. In Case 2 the identity sum_{j∈A} q u^(j,eps)_r = 1-s is correct, but the reuse of q makes the proof harder to follow; consider renaming the Case 2 quantity, e.g. q_u.
  3. [Proposition 2.7] The proof of Proposition 2.7 in Appendix A.2 invokes Step 1 of Theorem 5.1, a result proved later in the paper. A forward-reference note in Section 2 would help the reader.
  4. [General] Minor typographical issues: the overline on eBH is lost in several places (e.g., Definition 3.1 and elsewhere), and the displayed definitions of f_A and F_A in Theorem 5.1 could be numbered for easier reference.

Circularity Check

0 steps flagged

No significant circularity: the complete-class and admissibility theorems are derived from external representation lemmas, not from the conclusions they establish; the omitted Clerico adaptation is a proof gap, not an input-output identity.

full rationale

The derivation chain is not circular. The central complete-class claim (Theorem 5.1) is established by deriving, for an arbitrary D in SP_alpha, a weighted-mean eBH procedure that dominates it. The load-bearing Step 1 uses an external functional-analytic representation lemma attributed to Clerico (2026) to convert the FDR constraint E[f_A(X,Z(X))] <= 1 into domination by a weighted-mean function; the target admissibility result is not an assumption of that lemma. The weights of the weighted-mean class are not fitted to FDR data, and the simulation's learned weights are illustrative and do not feed back into the theorems. The paper does rely on self-cited external results, most notably 'The functions M_lambda are the only admissible e-merging functions for arbitrary e-values (Wang, 2025)', but this characterization is used for motivation and definitional context, not as a premise of Theorem 5.1's proof. The only manuscript-flagged gap is in Theorem 5.1, Step 1: the representation lemma is invoked with 'the same proof idea applies after a simple adjustment, and we omit the details'. This is an omitted proof / correctness risk, not circularity, because the lemma is independent of the FDR admissibility conclusion. The forward reference in Proposition 2.7's proof to Step 1 of Theorem 5.1 is a transparent dependency, not a cycle: Step 1 is proved independently and does not rely on Proposition 2.7. No fitted parameter is renamed as a prediction, and no admissible element is defined in terms of the class it is claimed to characterize. Therefore the paper exhibits no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The theory is parameter-free: it quantifies over all lambda, alpha, and K and derives admissibility/complete-class statements. No constants are fitted to data in the proofs. The simulation uses a training split to choose lambda and fixes eta = 0.45, but these choices do not enter the theorems. The axioms above are the unproved inputs the results rely on.

axioms (3)
  • standard math Axiom of choice / Zorn's lemma, so maximal elements and chain upper bounds can be selected.
    Explicitly assumed in Section 2 ('we assume the axiom of choice') and used in Theorem 5.1 Step 2 and Theorem 5.3 to select maximal elements.
  • domain assumption E-merging dominance lemma: a bounded F with E[F(X)] <= 1 under coordinatewise e-value constraints is dominated by a weighted mean (Clerico 2026, adapted; Wang 2025 characterization).
    Invoked without full proof in Theorem 5.1 Step 1; this is load-bearing for representing arbitrary procedures as weighted-mean eBH procedures.
  • domain assumption Closed eBH built from any e-collection controls FDR at level alpha (Xu et al. 2026).
    Used to place weighted-mean eBH procedures in SP_alpha and PP_alpha around Definition 3.1.

reviewed 2026-08-02 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Admissibility and Complete Classes for False Discovery Rate Control with E-values." pith.science (2026). https://pith.science/paper/SGQFVK2B

@misc{pith2026260714380,
  author       = {Pith},
  title        = {Pith review of: Admissibility and Complete Classes for False Discovery Rate Control with E-values},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SGQFVK2B}},
  note         = {Machine review of arXiv:2607.14380}
}
Share X Bluesky LinkedIn Reddit HN
abstract

The false discovery rate (FDR) is the most widely used error metric in modern multiple testing. We provide the first comprehensive analysis of the admissibility of e-value-based procedures with FDR control. We consider both simultaneous and point procedures and introduce strong and weak notions of dominance. We show that every simultaneous procedure is strongly, and hence weakly, dominated by an admissible weighted-mean closed e-Benjamini-Hochberg ($\overline{\mathrm{eBH}}$) procedure, so weighted-mean $\overline{\mathrm{eBH}}$ procedures form a complete class. Moreover, every constant-free weighted-mean $\overline{\mathrm{eBH}}$ procedure is admissible at every level. Within the symmetric class, the usual mean $\overline{\mathrm{eBH}}$ procedure is the largest element if and only if the FDR level is small enough; otherwise this class has no largest element. We also obtain results on the admissibility of symmetric $\overline{\mathrm{eBH}}$ procedures with non-zero constant terms, and give guidance on the choice of the constant terms. Point e-testing procedures have a parallel theory for admissibility, where point weighted-mean $\overline{\mathrm{eBH}}$ procedures form a complete class. These results highlight the central role of weighted-mean $\overline{\mathrm{eBH}}$ procedures in multiple testing.

Figures

Figures reproduced from arXiv: 2607.14380 by Liulei Sun, Ruodu Wang.

Figure 1
Figure 1. Figure 1: Empirical TPR for three procedures. 1. Scale distortion. Replace the training data by (1 + δ)Xt,i, where δ ∈ {0.25, 0.5}. 2. White noise perturbation. Replace the training data by Xt,i+σεt,i, where the independent noise variables εt,i ∼ N (0, 1) and σ ∈ {0.5, 1}. 3. Rank-breaking permutation. Draw C uniformly from all subsets of K having cardinality K/2, and, conditional on C, draw an independent per￾mutat… view at source ↗
Figure 2
Figure 2. Figure 2: Empirical TPR for different types of training data contamination. [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

20 extracted references · 3 linked inside Pith

  1. [1]

    Barber, R. F. and Ramdas, A. (2017). The p-filter: Multilayer false discovery rate control for grouped hypotheses.Journal of the Royal Statistical Society Series B: Statistical Methodology,79(4), 1247–1268

    2017

  2. [2]

    and Hochberg, Y

    Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing.Journal of the Royal Statistical Society Series B,57(1), 289–300

    1995

  3. [3]

    and Yekutieli, D

    Benjamini, Y. and Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency.Annals of Statistics,29(4), 1165–1188

    2001

  4. [4]

    and Roquain, E

    Blanchard, G. and Roquain, E. (2008). Two simple sufficient conditions for FDR control.Electronic Journal of Statistics,2, 963–992

    2008

  5. [5]

    Clerico, E. (2026). A simple geometric proof for the characterisation of e-merging functions.Statistics & Probability Letters,236, 110750

    2026

  6. [6]

    Goeman, J. (2026). A uniform improvement of the Benjamini-Hochberg proce- dure via e-Closure.arXiv:2606.01854

    Pith/arXiv arXiv 2026

  7. [7]

    J., Hemerik, J

    Goeman, J. J., Hemerik, J. and Solari, A. (2021). Only closed testing procedures are admissible for controlling false discovery proportions.Annals of Statistics, 49(2), 1218–1238. Gr¨ unwald, P., de Heide, R. and Koolen, W. M. (2024). Safe testing.Journal of the Royal Statistical Society Series B,86(5), 1091–1128

    2021

  8. [8]

    and Ramdas, A

    Ignatiadis, N., Wang, R. and Ramdas, A. (2024). E-values as unnormalized weights in multiple testing.Biometrika,111(2), 417–439

    2024

  9. [9]

    and Ramdas, A

    Ignatiadis, N., Wang, R. and Ramdas, A. (2026). Tiny but uniform improve- ments of adaptive BH procedures via compound e-values.arXiv:2603.21424

    arXiv 2026

  10. [10]

    and Ren, Z

    Lee, J. and Ren, Z. (2024). Boosting e-BH via conditional calibration. arXiv:2404.17562

    Pith/arXiv arXiv 2024

  11. [11]

    and Wang, R

    Ramdas, A. and Wang, R. (2025). Hypothesis testing with e-values.Foundations and Trends in Statistics,1(1–2), 1–390

    2025

  12. [12]

    and Barber, R

    Ren, Z. and Barber, R. F. (2024). Derandomised knockoffs: leveraging e-values for false discovery rate control.Journal of the Royal Statistical Society Series B,86(1), 122–154. 27

    2024

  13. [13]

    Shafer, G. (2021). Testing by betting: A strategy for statistical and scientific communication.Journal of the Royal Statistical Society, Series A, 184(2), 407–431

    2021

  14. [14]

    and Goeman, J

    Solari, A. and Goeman, J. J. (2017). Minimally adaptive BH: A tiny but uni- form improvement of the procedure of Benjamini and Hochberg.Biometrical Journal,59(4), 776–780

    2017

  15. [15]

    Storey, J. D. (2002). A direct approach to false discovery rates.Journal of the Royal Statistical Society Series B: Statistical Methodology,64(3), 479–498

    2002

  16. [16]

    and Wang, R

    Vovk, V., Wang, B. and Wang, R. (2022). Admissible ways of merging p-values under arbitrary dependence.Annals of Statistics,50(1), 351–375

    2022

  17. [17]

    and Wang, R

    Vovk, V. and Wang, R. (2024). True and false discoveries with independent and sequential e-values.Canadian Journal of Statistics,52(4), e11833

    2024

  18. [18]

    Wang, R. (2025). The only admissible way of merging arbitrary e-values. Biometrika,112(2), asaf020

    2025

  19. [19]

    and Ramdas, A

    Wang, R. and Ramdas, A. (2022). False discovery rate control with e-values. Journal of the Royal Statistical Society Series B,84(3), 822–852

    2022

  20. [20]

    Admissibility and Complete Classes for False Discovery Rate Control with E-values

    Xu, Z., Solari, A., Fischer, L., de Heide, R., Ramdas, A. and Goeman, J. (2026). Bringing closure to false discovery rate control: A general principle for multiple testing.arXiv:2509.02517. 28 Supplementary Material for “Admissibility and Complete Classes for False Discovery Rate Control with E-values” This supplement contains some proofs and counterexamp...

    Pith/arXiv arXiv 2026

This paper was first reviewed by deepseek-v4-flash on August 2, 2026.