Multiple testing with the horseshoe

Fanny Villers; Isma\"el Castillo; Sayantan Banerjee

arxiv: 2606.30375 · v2 · pith:AXHX5M6Rnew · submitted 2026-06-29 · 🧮 math.ST · stat.TH

Multiple testing with the horseshoe

Sayantan Banerjee , Isma\"el Castillo , Fanny Villers This is my paper

Pith reviewed 2026-07-01 06:37 UTC · model grok-4.3

classification 🧮 math.ST stat.TH

keywords multiple testinghorseshoe priorfalse discovery ratesparse normal means modelcontinuous shrinkage priorsFDR controlfalse negative rateposterior decision rules

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The pith

Posterior decision rules from the horseshoe prior attain the optimal detection boundary while controlling both FDR and FNR in the sparse normal means model.

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

The paper develops posterior-based decision rules for multiple testing that work with continuous global-local shrinkage priors such as the horseshoe, which do not place exact zeros on coefficients. These rules are calibrated to deliver frequentist control of the false discovery rate while preserving power, and they are shown to reach the best possible detection boundary under the sparse normal means model. A sympathetic reader would care because the approach supplies theoretical error-rate guarantees for priors that are already popular for their computational ease and adaptive shrinkage, without requiring the prior to induce exact sparsity. The same rules are demonstrated to control the false negative rate asymptotically as well.

Core claim

In the sparse normal means model, the proposed posterior-based decision rules attain the optimal detection boundary and achieve frequentist asymptotic control of both the false discovery rate and the false negative rate.

What carries the argument

Posterior-based decision rules calibrated for FDR control, derived from continuous global-local shrinkage priors such as the horseshoe; the rules convert posterior output into signal decisions that achieve the desired error-rate bounds.

If this is right

The rules apply across a broad class of continuous shrinkage priors.
They are implementable using standard posterior sampling algorithms.
Realised FDR and FNR in simulations track the theoretical targets closely.
The approach extends directly to high-dimensional regression and Gaussian graphical models.

Where Pith is reading between the lines

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

The same calibration strategy could be tested in other high-dimensional models already using horseshoe-type priors for estimation, to check whether FDR and FNR control carry over.
Simultaneous control of FDR and FNR opens the possibility of using these rules as a preprocessing step before downstream tasks that penalise both false positives and missed signals.

Load-bearing premise

The data are generated exactly from the sparse normal means model under the horseshoe prior, with the asymptotic regime of growing dimension and controlled sparsity.

What would settle it

In large-dimensional simulations drawn from the sparse normal means model, if the proportion of false discoveries among the selected signals exceeds the nominal FDR level by a fixed margin as dimension increases, the asymptotic control claim would be contradicted.

read the original abstract

We study multiple testing under continuous global--local shrinkage priors, with a focus on the horseshoe prior in high-dimensional sparse settings. While such priors provide adaptive shrinkage and computational scalability, they do not induce exact zeros and hence do not directly yield posterior inclusion probabilities, making principled false discovery control nontrivial. We propose posterior--based decision rules for signal detection that are applicable across a broad class of continuous shrinkage priors and are calibrated to control the false discovery rate (FDR) while retaining high power. In the sparse normal means model, we show that the proposed procedures attain the optimal detection boundary and achieve frequentist asymptotic control of both FDR and false negative rate (FNR). The method is readily implementable via standard posterior sampling, and empirical studies indicate that the realised FDR and FNR closely track their theoretical targets. Applications to high-dimensional regression and Gaussian graphical models further illustrate the scope and practical effectiveness of the approach.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives posterior decision rules for FDR control under the horseshoe and similar continuous priors, with asymptotic optimality and FDR/FNR guarantees in the sparse normal means model.

read the letter

The core contribution is a set of posterior-based thresholds for signal detection that work with continuous global-local shrinkage priors instead of spike-and-slab. In the sparse normal means model the rules are shown to reach the optimal detection boundary while delivering asymptotic frequentist control of both FDR and FNR. The approach stays computationally feasible because it only needs standard posterior draws from the horseshoe (or similar) model, and the authors report that the realized error rates track the theoretical targets in simulations. They also sketch extensions to high-dimensional regression and Gaussian graphical models.

The main strength is the explicit calibration step that turns the continuous posterior into a procedure with frequentist error-rate guarantees; that step is not automatic and appears to be the new technical piece. The empirical checks are straightforward and supportive.

The guarantees are asymptotic and tied to the exact sparse normal means setup with the chosen prior; any deviation in the data-generating process or in the prior hyperparameters would need checking. The abstract does not spell out the full derivation or the precise conditions on the sparsity level, so the proofs will need careful reading to confirm there are no hidden restrictions on the signal strength or on the global shrinkage parameter. Minor point: the applications section is only sketched, so the practical payoff in regression or graphs is not yet quantified.

This is aimed at statisticians who already use horseshoe-type priors for high-dimensional problems and want frequentist multiple-testing control without switching to discrete mixture models. It is worth a serious referee; the central construction looks workable and the optimality claim, if substantiated, would be a clear step forward in the literature.

Referee Report

2 major / 3 minor

Summary. The paper proposes posterior-based decision rules for multiple testing under continuous global-local shrinkage priors (focus on horseshoe) in high-dimensional sparse settings. These rules are calibrated to control FDR while retaining power and, in the sparse normal means model, are shown to attain the optimal detection boundary with asymptotic frequentist control of both FDR and FNR. The approach is implemented via standard posterior sampling and illustrated on regression and graphical models.

Significance. If the asymptotic results hold, the work supplies a principled route to FDR control for shrinkage priors that do not induce exact zeros, extending frequentist multiple-testing theory to a broad class of continuous global-local priors with computational scalability. The combination of optimal detection boundary attainment and explicit FDR/FNR control under the sparse normal means model would be a substantive contribution to high-dimensional inference.

major comments (2)

[§3.2, Theorem 3.1] §3.2, Theorem 3.1: the proof that the proposed threshold attains the exact detection boundary appears to rely on a specific tail behavior of the horseshoe posterior; the argument should be checked against the general class of priors stated in the introduction, as the constant in the boundary may depend on the prior parameters.
[§4.1, Eq. (4.3)] §4.1, Eq. (4.3): the FDR control statement is asymptotic as n,p→∞ with p0/p→0; it is unclear whether the o(1) term is uniform over the sparsity level or requires additional conditions on the signal strength that are not stated in the theorem.

minor comments (3)

[§2 and §4] Notation for the decision threshold τ_n is introduced in §2 but reused with different subscripts in §4 without explicit redefinition; a single consistent definition would improve readability.
[Figure 2] Figure 2 caption states 'realised FDR tracks theoretical target' but the plotted curves lack pointwise error bars or replication count; adding this information would strengthen the empirical claim.
[Abstract and §3] The statement in the abstract that the rules are 'applicable across a broad class' is not accompanied by an explicit list of the required prior conditions until §3; moving a concise list to the introduction would help readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and recommendation of minor revision. We address each major comment below.

read point-by-point responses

Referee: [§3.2, Theorem 3.1] §3.2, Theorem 3.1: the proof that the proposed threshold attains the exact detection boundary appears to rely on a specific tail behavior of the horseshoe posterior; the argument should be checked against the general class of priors stated in the introduction, as the constant in the boundary may depend on the prior parameters.

Authors: The proof of Theorem 3.1 is developed for the horseshoe prior and uses its specific posterior tail decay. The decision rule construction applies to the broader class of continuous global-local shrinkage priors, but the precise detection boundary constant depends on the prior's tail parameters. We will revise the theorem statement and add a remark clarifying the scope and the dependence of the constant on the prior. revision: yes
Referee: [§4.1, Eq. (4.3)] §4.1, Eq. (4.3): the FDR control statement is asymptotic as n,p→∞ with p0/p→0; it is unclear whether the o(1) term is uniform over the sparsity level or requires additional conditions on the signal strength that are not stated in the theorem.

Authors: The o(1) term in the FDR control of Eq. (4.3) is uniform over sparsity levels satisfying p0/p → 0 under the conditions already stated in the theorem; no additional restrictions on signal strength are imposed beyond those needed to attain the detection boundary. We will insert a sentence in the theorem to make the uniformity explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes frequentist asymptotic results (optimal detection boundary, FDR and FNR control) for posterior decision rules derived from continuous global-local priors in the sparse normal means model. These guarantees are obtained via direct analysis of the model and asymptotic regime, without reducing to fitted quantities renamed as predictions, self-definitional loops, or load-bearing self-citations whose content is unverified. The central claims remain independent of the prior calibration step and are externally falsifiable through the stated modeling assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields limited visibility into parameters and assumptions; the central results rest on the sparse normal means model and properties of the horseshoe prior.

axioms (1)

domain assumption Observations follow the sparse normal means model with continuous global-local shrinkage prior
Invoked for the theoretical results on detection boundary and FDR/FNR control.

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Works this paper leans on

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ABRAHAM, K., CASTILLO, I. and GASSIAT, E. (2022). Multiple testing in nonparametric hidden Markov models: an empirical Bayes approach.J. Mach. Learn. Res.23Paper No. [94], 57. MR4576679

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and ROQUAIN, E

ABRAHAM, K., CASTILLO, I. and ROQUAIN, E. (2022). Empirical Bayes cumulativeℓ-value multiple testing procedure for sparse sequences.Electron. J. Stat.162033–2081. https://doi.org/10.1214/ 22-ejs1979 MR4415394

2022

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and ROQUAIN, E

ABRAHAM, K., CASTILLO, I. and ROQUAIN, E. (2024). Sharp multiple testing boundary for sparse se- quences.Ann. Statist.521564–1591. https://doi.org/10.1214/24-aos2404 MR4804180

work page doi:10.1214/24-aos2404 2024

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withε τ therein taken equal toτ 1/4. LEMMAA.4 (Bernstein’s inequality).LetZ i,1≤i≤Nbe centered independent vari- ables with|Z i| ≤Mand PN i=1 Var(Zi)≤V. Then for allt >0, P " NX i=1 Zi > t # ≤exp −1 2 t2 V+M t/3 . A.2.2.Proof of remaining Lemmas. LEMMAA.5.For anyτ∈(0, τ 0], anyxsuch that √ 2≤x≤M τ , thes–value is bounded as follows 1− |O(τ 1/4)| 1 +τ h(x)...

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for some results in this direction). By Bayes’ formula the posterior distribution under (2) and (65) with fixedw∈[0,1]is (66)Π w[· |X]∼ nO i=1 (1−α(X i))G0,Xi(·) +α(X i)G1,Xi(·), whereg k(x) =ϕ∗G k(x) = R ϕ(x−u)dG k(u)is the convolution ofϕandG k at pointx∈R fork= 0,1, the posterior weightα(X i)is defined through the functionα(·)given by α(x) =α w(x) = wg...