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REVIEW 3 major objections 5 minor 43 references

Robust Bayesian high-dimensional variable selection and inference with the horseshoe family of priors

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Using a Laplace working likelihood and Gibbs samplers, the paper develops robust Bayesian regression with horseshoe, horseshoe+, and regularized horseshoe priors and reports that the one-group priors can yield calibrated marginal credible…

desk verdict Useful new Gibbs samplers and a serious simulation study, but the abstract overclaims family-level valid inference: regularized horseshoe undercovers badly (0.649) in the paper's own tables. read the letter →

arxiv 2507.10975 v2 pith:SP7IEC25 submitted 2025-07-15 stat.ME

Kun Fan , Srijana Subedi , Vishmi Ridmika Dissanayake Pathiranage , Cen Wu This is my paper · ORCID
classification stat.ME MSC 62F1562J0762F35
keywords horseshoepriorhorseshoe+regularizedrobustBayesianvariableselectionLaplaceworkinglikelihoodGibbssamplinghigh-dimensionalcredibleintervalsheavy-tailederrors
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks whether one-group horseshoe-style priors, which never force coefficients exactly to zero, can support both variable selection and trustworthy interval inference in robust high-dimensional regression. It develops robust Bayesian regression models that pair a Laplace working likelihood with horseshoe, horseshoe+, and regularized horseshoe priors, and derives Gibbs samplers for each. In simulations with heavy-tailed and skewed errors, the proposed robust methods outperform their non-robust counterparts in estimation, selection, and prediction. The central empirical claim is that, despite lacking exact sparsity, the horseshoe and horseshoe+ versions yield 95% credible intervals with near-nominal coverage under p > n, while the regularized-horseshoe version under-covers in some settings.

What carries the argument

The load-bearing mechanism is the Laplace likelihood's representation as a normal-exponential scale mixture, combined with the auxiliary-mixture representation of half-Cauchy priors as inverse-gamma scale mixtures. This turns the full posterior into a tractable Gibbs sampler where each coefficient's full conditional is normal, with a shrinkage factor $\kappa_j = (1 + \lambda^2 s_j^2 \sum_i x_{ij}^2 / (\tau^{-1} \xi^2 \tilde v_i))^{-1}$ governing how much the posterior mean is pulled toward zero. Propositions 1.1 through 1.3 express the posterior mean as $(1 - \kappa_j)$ times a least-squares-like update and give the density of $\kappa_j$ for horseshoe and horseshoe+, which is how the paper connects one-group shrinkage to feature selection.

What would settle it

A decisive check is to recompute the empirical coverage of the 95% credible interval for the largest nonzero coefficient under t(2) errors with n = 200 and p = 1000, but with covariates drawn from an independent or banded correlation matrix instead of AR(1); the paper's claim predicts coverage near 0.95, and a coverage below 0.90 for RBHS would falsify the general claim.

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Extended reading notes

Core claim

The paper's central claim is that exact sparsity is not necessary for valid Bayesian inference in robust high-dimensional sparse regression. Under a Laplace working likelihood for the errors, horseshoe-family priors shrink noise coefficients so aggressively that effective sparsity emerges, and the resulting marginal credible intervals for the nonzero coefficients achieve empirical coverage near the nominal 95% level. The paper supports this with simulations at (n,p) = (100,500) and (200,1000), reporting that RBHS and RBHS+ maintain coverage roughly from 0.93 to 0.97 under normal and t(2) errors, while RBRHS falls to values such as 0.649 for the largest coefficient in the (200,1000) setting.

Load-bearing premise

The load-bearing premise is that a Laplace working likelihood paired with a horseshoe prior produces posterior credible intervals that hit their nominal coverage when there are more predictors than observations, even though published work cited by the paper shows a variance correction is needed for that likelihood in low dimensions and no correction is available in high dimensions.

Editorial extensions

If this is right

  • RBHS and RBHS+ provide a computationally cheap robust alternative to spike-and-slab models: 10,000 Gibbs iterations complete in seconds, with calibration maintained when the number of predictors far exceeds the sample size.
  • Under heavy-tailed t(2) errors, the robust horseshoe methods keep coverage near nominal while non-robust counterparts drop to roughly 0.7 to 0.8, so the robustness of the likelihood is what restores interval validity.
  • The proposed samplers are orders of magnitude faster than slice-sampling horseshoe quantile regression, making large-scale eQTL-style analyses practical.
  • In the rat eye eQTL case study, robust horseshoe models select fewer genes and achieve lower prediction error than non-robust versions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the calibration finding extends beyond AR(1) designs, robust high-dimensional inference could drop two-group spike-and-slab priors entirely, removing the need to tune a mixture indicator and lowering computational cost.
  • The paper's own RBRHS coverage failures suggest that adding a Gaussian ridge-like component to a horseshoe prior, while helpful for correlated predictors, can break interval calibration; testing whether the b prior or the extra shrinkage layer is responsible would be a direct follow-up.
  • A formal theory showing when continuous global-local priors yield calibrated credible intervals under a working likelihood, analogous to the spike-and-slab results, is the natural next step and would turn the simulation finding into a theorem.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper develops robust Bayesian regression models using the horseshoe, horseshoe+, and regularized horseshoe priors under a Laplace working likelihood, with explicit Gibbs samplers provided in Appendix C. It evaluates variable selection, estimation, credible interval coverage, computational speed, and a real eQTL application, and claims that the one-group horseshoe family can yield valid Bayesian credible intervals in high-dimensional robust regression even without exact sparsity.

Significance. If the family-level inference claim were established, this would be a useful counterpoint to the view that exact sparsity is necessary for valid high-dimensional Bayesian inference, and the explicit Gibbs samplers would be a practical contribution over slice sampling. The simulation infrastructure is solid: the coverage study uses 1,000 replicates, a DGP matching Fan et al. (2024) for cross-comparison, and external comparators such as HSBQR and Bayesreg. The paper is also honest about the absence of a variance-correction theorem for p>n. However, the paper's own results do not support the family-level inference claim: RBRHS coverage drops to 0.649 in Table 15, so the evidence supports RBHS and RBHS+ in a narrow AR(1) design rather than the whole horseshoe family.

major comments (3)
  1. [Abstract; §3 (Table 4); Appendix A.2 (Table 15)] The abstract and Section 5 claim that 'one-group horseshoe priors' yield valid credible intervals even without exact sparsity, but the coverage tables include RBRHS, which is part of that family. In Table 4, RBRHS coverage for β3 is 0.853 under N(0,1) and 0.805 under t(2); in Table 15 these are 0.781 and 0.649, while RBHS and RBHS+ remain at 0.923–0.971. The family-level statement is therefore contradicted by the paper's own evidence; please restrict the validity claim to RBHS and RBHS+, or provide a method-specific calibration argument for RBRHS, and revise the abstract, contribution list, and Discussion accordingly.
  2. [§2.2 (Yang et al. (2016) remark; §3 high-dimensional inference)] The paper acknowledges that Yang et al. (2016) requires a posterior variance correction for the Laplace working likelihood and that this correction is unavailable when p>n, and it then relies on the horseshoe prior to repair calibration. No theorem, finite-sample bound, or asymptotic argument is supplied to replace the correction. Given that RBRHS fails the coverage check, the inference claim rests on empirical calibration in one AR(1) design with three strong signals. Please either provide theoretical conditions under which the resulting intervals are calibrated, or explicitly reframe the inference conclusions as empirical and scope them to the settings simulated.
  3. [Appendix C.1.3 (and C.2.3)] The inverse-Gaussian update for \tilde v_i appears inconsistent with the completed-square derivation. For C.1.3, the displayed conditional has exponent −τ\tilde v_i − τ r_i²/(2ξ² \tilde v_i); under the standard inverse-Gaussian parameterization, the stated parameters μ=√(2ξ²/r_i²), λ=2τ give exponent −τ r_i² \tilde v_i/(2ξ²) − τ/\tilde v_i, which has the roles of \tilde v_i and 1/\tilde v_i swapped. Please check whether the displayed sampler is a reciprocal transformation or a typo; as written, the sampler does not target the stated full conditional. The same issue appears in C.2.3.
minor comments (5)
  1. [§3 and Table 4] The notation β1 and β2 is used both for scalar coefficients and for the nonzero/zero blocks, and Table 4's 'Coverage of β2' row actually refers to zero coefficients, not coefficient β2=1.5; please use unambiguous notation.
  2. [Tables 4 and 17] Tables 4 and 17 report the same (n,p)=(100,500), AR(1) setting but give different coverage for the proposed methods (e.g., RBHS β2: 0.940 vs 0.924); please clarify whether these are independent simulation runs or correct the inconsistency.
  3. [Throughout] There are repeated typos, including 'repametrization', 'an Bayesian', 'multicolinearity', 'proportions' for 'propositions', and 'dintinct meachnisms'; a careful proofread is needed.
  4. [§3, after Table 9] The text says false positives are 'very low, often close to zero', but for the log-normal non-i.i.d. setting all six methods report about one false positive on average; please adjust the summary sentence to match the table.
  5. [Main text and Appendix numbering] Appendix A.2 and Table 15 are referenced as 'Figures 5 and 6 and Table 15 in the Appendix,' and the appendix section titles should be checked for consistency with the actual table and figure numbering.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the credible-interval claim is an empirical calibration result, and self-citations are comparative rather than load-bearing.

full rationale

The paper's derivation chain is self-contained: it specifies a hierarchical model with a Laplace working likelihood and horseshoe-family priors, derives full conditional distributions and Gibbs samplers (Appendix C), runs MCMC, and evaluates 95% empirical coverage on 1000 simulated datasets (Tables 4 and 15). No parameter is fitted to force the coverage probabilities, no coverage target enters the prior specification, and the benchmarks include external methods (HSBQR, bayesreg). The load-bearing external reference, Yang et al. (2016), is used only to state that the Laplace working likelihood needs a variance correction in low dimensions and that the correction is unavailable when p > n; the paper then tests the horseshoe priors empirically without claiming a theorem. Self-citations (Fan et al. 2024; Ren et al. 2023; Liu et al. 2024) are motivational or are used to adopt the same simulation DGP for cross-comparison, which is a legitimate comparative choice and not a reduction of the conclusion to the input. The internal inconsistency that RBRHS undercovers badly (0.649 in Table 15) while the abstract attributes valid inference to the whole horseshoe family is an overgeneralization/correctness risk, not a circular step, because the reported coverage numbers are not constructed from the conclusion. Overall, no equation in the paper reduces to its own inputs, and no prediction is forced by a fitted parameter or a self-citation chain.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

All derivations rely on two standard mixture identities plus a domain assumption about the behavior of the Laplace working likelihood in high dimensions. The central inference claim is not derived but empirically demonstrated; the regularized horseshoe results show the domain assumption is not guaranteed across the family.

free parameters (3)
  • sigma_beta0^2 (intercept prior variance)
    Hyperparameter in all six hierarchical models (Section 2.2, Appendix C); no numerical value is reported in the paper, so exact replication of the simulation intervals is not possible without author contact.
  • e, f (Gamma hyperparameters for tau)
    Prior on the Laplace scale tau in RBHS/RBHS+/RBRHS; values not given in Section 2 or Appendix C; the posterior scale and credible interval widths depend on these.
  • c, d (inverse-gamma hyperparameters for b^2 in RBRHS)
    The regularized horseshoe's ridge parameter b^2 ~ IG(c/2, d/2) (Section 2.2.3) governs the extra Gaussian shrinkage; c and d are never specified for the simulations, and the under-coverage of RBRHS is plausibly driven by this choice. A sensitivity analysis is missing.
assumptions (3)
  • standard math Laplace errors admit the scale-mixture representation epsilon_i = tau^{-1} xi sqrt(v_i) z_i with v_i ~ Exp(1), z_i ~ N(0,1), xi = sqrt(8)
    Invoked in Section 2.1 to build the robust hierarchical model; this is the Kozumi-Kobayashi (2011) identity, a known mathematical result.
  • standard math A half-Cauchy prior on a scale parameter s is equivalent to s^2|nu ~ IG(1/2,1/nu), nu ~ IG(1/2,1)
    Used to derive the Gibbs samplers (Section 2.2.1 and Appendix C) following Wand et al. (2011) and Makalic and Schmidt (2015); a known auxiliary-mixture identity.
  • domain assumption The Laplace working likelihood yields calibrated posterior credible intervals in p > n problems once a shrinkage prior is used, even though Yang et al. (2016) showed a variance correction is required in low-dimensional Bayesian quantile regression
    This is the premise the entire inference claim rests on (Section 2.2); it is asserted from simulations only, and the paper's own RBRHS results (Tables 4, 15) violate it for one of the three priors.

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Pith. "Pith review of Robust Bayesian high-dimensional variable selection and inference with the horseshoe family of priors." pith.science (2026). https://pith.science/paper/SP7IEC25

@misc{pith2026250710975,
  author       = {Pith},
  title        = {Pith review of: Robust Bayesian high-dimensional variable selection and inference with the horseshoe family of priors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SP7IEC25}},
  note         = {Machine review of arXiv:2507.10975}
}
read the original abstract

Frequentist robust variable selection has been extensively investigated in high-dimensional regression. Despite success, developing the corresponding statistical inference procedures remains a challenging task. Recently, tackling this challenge from a Bayesian perspective has received much attention. In literature, the two-group spike-and-slab priors that can induce exact sparsity have been demonstrated to yield valid inference in robust sparse linear models. Nevertheless, another important category of sparse priors, the horseshoe family of priors, including horseshoe, horseshoe+, and regularized horseshoe priors, has not yet been examined in robust high-dimensional regression by far. Their performance in variable selection and especially statistical inference in the presence of heavy-tailed model errors is not well understood. In this paper, we address the question by developing robust Bayesian hierarchical models utilizing the horseshoe family of priors along with an efficient Gibbs sampling scheme. We show that compared with competing methods with alternative sampling strategies such as slice sampling, our proposals lead to superior performance in variable selection, Bayesian estimation and statistical inference. In particular, our numeric studies indicate that even without imposing exact sparsity, the one-group horseshoe priors can still yield valid Bayesian credible intervals under robust high-dimensional linear regression models. Applications of the proposed and alternative methods on real data further illustrates the advantage of the proposed methods.

Figures

Figures reproduced from arXiv: 2507.10975 by the authors.

Figure 1
Figure 1. F1 score, MCC score and Estimation error under AR(1) correlation, [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Credible intervals with empirical coverage probabilities under a 95% nominal level are shown for [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Credible intervals with empirical coverage probabilities under a 95% nominal level are shown for [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: F1 score, MCC score and Estimation error under AR(1) correlation, [PITH_FULL_IMAGE:figures/full_fig_p032_4.png]
Figure 5
Figure 5. Figure 5: Credible intervals with empirical coverage probabilities under a 95% nominal level for the first 10 [PITH_FULL_IMAGE:figures/full_fig_p039_5.png]
Figure 6
Figure 6. Figure 6: Credible intervals with empirical coverage probabilities under a 95% nominal level for the first 10 [PITH_FULL_IMAGE:figures/full_fig_p039_6.png]
Figure 7
Figure 7. Figure 7: Potential scale reduction factor (PSRF) against iterations for nonzero coefficients in simulation for [PITH_FULL_IMAGE:figures/full_fig_p040_7.png]
Figure 8
Figure 8. Figure 8: Potential scale reduction factor (PSRF) against iterations for nonzero coefficients in simulation for [PITH_FULL_IMAGE:figures/full_fig_p041_8.png]
Figure 9
Figure 9. Figure 9: Potential scale reduction factor (PSRF) against iterations for nonzero coefficients in simulation for [PITH_FULL_IMAGE:figures/full_fig_p042_9.png]

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