{"id":"23cda6cc-dfa2-4104-b594-0865672b9a75","arxiv_id":"2507.10975","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"New Gibbs samplers for robust median regression with horseshoe, horseshoe+, and regularized horseshoe priors; simulations show the first two yield near-95% credible intervals in high dimensions even without exact sparsity.","lead":"This paper builds fast Bayesian methods for finding important variables in high-dimensional regression when data contain outliers, using horseshoe-family priors with a robust Laplace error model. The simulations show near-nominal credible interval coverage for horseshoe and horseshoe+ priors, while regularized horseshoe under-covers badly in several settings.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central inference claim overreaches: RBRHS undercovers badly (0.649 in Table 15) and no theorem or variance correction replaces Yang et al. (2016), so the evidence supports only HS/HS+ in a narrow design, not the whole horseshoe family.","rationale":"The reader's verdict is CONDITIONAL, and I agree. The paper's explicit Gibbs samplers and large simulation study are genuine contributions, and the evidence for RBHS and RBHS+ calibration is reasonably strong within the tested settings. However, the abstract's unqualified statement about 'the one-group horseshoe priors' is contradicted by the paper's own RBRHS results, where coverage falls to 0.649 for beta3 under t(2) errors. The reader's weakest assumption captures the same underlying issue: valid calibration is prior-dependent and no theoretical replacement for the Yang et al. (2016) variance correction is supplied. My concrete test is designed to separate an intrinsic failure of RBRHS from a fixable hyperparameter artifact, which determines whether the family-level claim can be salvaged or must be narrowed.","tokens_in":51047,"tokens_out":5762,"duration_ms":73964,"concrete_test":"Rerun the Table 15 inference design (n = 200, p = 1000, AR(1), t(2), beta = (1, 1.5, 2, 0, ...)) with RBRHS using the same Gibbs sampler but with the regularized-horseshoe b2 prior replaced by (i) a diffuse inverse-gamma prior (shape = rate = 0.01) and (ii) b fixed to 10^6, which reduces RBRHS to RBHS. If beta3 coverage stays below 0.85 in both settings, the undercoverage is intrinsic to RBRHS and the family-level claim is false as written; if coverage recovers to about 0.95 only in setting (ii), the claim must be narrowed to HS/HS+ with RBRHS explicitly excluded.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's central claim is that one-group horseshoe priors can yield valid Bayesian credible intervals in high-dimensional robust regression without exact sparsity. The load-bearing premise, acknowledged in Section 2.2, is that the Laplace working likelihood can be used when p > n without the posterior variance-correction step that Yang et al. (2016) proved necessary in low dimensions; the horseshoe prior alone is assumed to repair calibration. This premise is not supported by any theorem, and the paper's own simulations are internally inconsistent with the family-level claim: in Table 15 (n = 200, p = 1000, t(2) errors), RBRHS has coverage 0.649 for beta3, while RBHS and RBHS+ give 0.971 and 0.968. The abstract and introduction attribute valid inference to 'the horseshoe family of priors' generally, but Tables 4 and 15 support only RBHS and RBHS+ under one AR(1) design with three strong signals. Thus the generalization to the full family, and to other designs or signal strengths, is the weakest load-bearing step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":51258,"tokens_out":15074,"duration_ms":170671,"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":[{"comment":"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.","section":"Abstract; §3 (Table 4); Appendix A.2 (Table 15)"},{"comment":"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.","section":"§2.2 (Yang et al. (2016) remark; §3 high-dimensional inference)"},{"comment":"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.","section":"Appendix C.1.3 (and C.2.3)"}],"minor_comments":[{"comment":"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.","section":"§3 and Table 4"},{"comment":"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.","section":"Tables 4 and 17"},{"comment":"There are repeated typos, including 'repametrization', 'an Bayesian', 'multicolinearity', 'proportions' for 'propositions', and 'dintinct meachnisms'; a careful proofread is needed.","section":"Throughout"},{"comment":"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.","section":"§3, after Table 9"},{"comment":"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.","section":"Main text and Appendix numbering"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, read this one for the Gibbs samplers, not the abstract. They give explicit Gibbs samplers for robust median regression with the horseshoe, horseshoe+, and regularized horseshoe priors using a Laplace working likelihood, and they run a serious simulation study. That part is new and useful: Kohns and Szendrei used slice sampling and never looked at variable selection or coverage; this paper does, with 1,000 replicates under AR(1) and banded designs, heavy-tailed and heteroscedastic errors, and comparisons against HSBQR, Bayesreg, and non-robust counterparts. The speed advantage over slice sampling is dramatic and believable (Table 18). The Gibbs samplers in Appendix C are standard but internally consistent, and the coverage study is well structured for cross-comparison with the spike-and-slab literature.\n\nThe soft spot is the abstract's claim that \"the one-group horseshoe priors can still yield valid Bayesian credible intervals.\" The paper's own tables don't support that for the regularized horseshoe: RBRHS coverage drops to 0.853 (Table 4) and 0.649 (Table 15, t(2), beta3). The evidence supports RBHS and RBHS+ under one AR(1) design with three strong signals. The authors themselves note the failure and attribute it to the elastic-net connection, but the family-level claim in the abstract and introduction overreaches and should be narrowed.\n\nThe second soft spot is theoretical. They cite Yang et al. (2016) on the need for a variance correction for the Laplace working likelihood in low dimensions, then simply drop the correction in p > n and rely on the horseshoe prior to fix calibration. That might be true, but no theorem or adapted correction is offered; the claim rests entirely on a narrow simulation design. A serious reader will want either a theoretical argument or a much wider design sweep before accepting exact-sparsity-free inference as a general phenomenon.\n\nOn the mechanics: hyperparameters (sigma_beta0^2, e, f, c, d) are never reported; the code is promised for pqrBayes but not shipped; and Propositions 1.1-1.3 are printed with garbled formulas in the current PDF, which makes the shrinkage factor derivations hard to check. These are fixable in revision. Minor table typos exist too.\n\nBottom line: useful contribution, a real methods paper, and worth refereeing. But the inference claim needs to be narrowed or substantially reinforced, and the reproducibility issues need to be fixed before publication.","headline":"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.","tokens_in":51853,"tokens_out":3039,"would_cite":true,"duration_ms":33481,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F15","62J07","62F35"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["horseshoe prior","horseshoe+ prior","regularized horseshoe prior","robust Bayesian variable selection","Laplace working likelihood","Gibbs sampling","high-dimensional credible intervals","heavy-tailed errors"],"falsifier":"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.","tokens_in":50791,"feed_emoji":"📊","tokens_out":6499,"duration_ms":76663,"temperature":0.7,"pith_summary":"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.","feed_headline":"No exact sparsity needed for robust Bayesian intervals","feed_subtitle":"Gibbs samplers show one-group horseshoe priors keep ~95% coverage under t(2) errors and p>n.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the horseshoe prior and the global-local shrinkage mechanism used in RBHS.","marker":"[8]"},{"why":"Supplies the horseshoe+ prior and the extra local shrinkage layer used in RBHS+.","marker":"[13]"},{"why":"Supplies the regularized horseshoe prior and the inverse-gamma recommendation for b^2 used in RBRHS.","marker":"[14]"},{"why":"Provides the normal-exponential mixture representation of the Laplace distribution that underlies the robust Gibbs samplers.","marker":"[17]"},{"why":"Provides the auxiliary-variable scale-mixture representation of half-Cauchy distributions used to build the Gibbs updates.","marker":"[18]"},{"why":"Provides the simple Gibbs sampler for the horseshoe estimator that the robust samplers extend.","marker":"[19]"},{"why":"Documents the need for a posterior variance adjustment under the Laplace working likelihood, the obstacle the paper claims the horseshoe prior overcomes in high dimensions.","marker":"[26]"},{"why":"Shows horseshoe credible intervals can be calibrated in sparse normal means models, motivating the extension to robust regression.","marker":"[12]"},{"why":"Proposes the spike-and-slab robust Bayesian LASSO whose exact-sparsity inference results are the benchmark and whose simulation design is reused.","marker":"[3]"},{"why":"Proposes slice-sampling horseshoe Bayesian quantile regression, the main competing sampler the Gibbs approach is compared against.","marker":"[22]"}],"fun_headline_variants":["One-group horseshoe keeps ~95% coverage without exact sparsity","Horseshoe priors deliver robust credible intervals sans exact sparsity","Robust high-dim inference works with horseshoe, no exact sparsity needed","Bayesian intervals stay valid without exact sparsity in robust regression"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["One-group horseshoe keeps ~95% coverage without exact sparsity","Horseshoe priors deliver robust credible intervals sans exact sparsity","Robust high-dim inference works with horseshoe, no exact sparsity needed","Bayesian intervals stay valid without exact sparsity in robust regression"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001039,"raw_usage":{"total_tokens":4363,"prompt_tokens":929,"completion_tokens":3434,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":545,"completion_tokens_details":{"reasoning_tokens":3358}},"tokens_in":545,"tokens_out":3434,"duration_ms":26999,"temperature":1.0,"reasoning_tokens":3358,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:21:53.815092+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"The horseshoe estimator for sparse signals,","cited_arxiv_id":null,"evidence_quote":"Supplies the horseshoe prior and the global-local shrinkage mechanism used in RBHS."},{"cited_title":"The horseshoe+ estimator of ultra-sparse signals,","cited_arxiv_id":null,"evidence_quote":"Supplies the horseshoe+ prior and the extra local shrinkage layer used in RBHS+."},{"cited_title":"Sparsity information and regularization in the horseshoe and other shrinkage priors,","cited_arxiv_id":null,"evidence_quote":"Supplies the regularized horseshoe prior and the inverse-gamma recommendation for b^2 used in RBRHS."},{"cited_title":"Gibbs sampling methods for bayesian quantile regres- sion,","cited_arxiv_id":null,"evidence_quote":"Provides the normal-exponential mixture representation of the Laplace distribution that underlies the robust Gibbs samplers."},{"cited_title":"Mean field variational bayes for elaborate distributions,","cited_arxiv_id":null,"evidence_quote":"Provides the auxiliary-variable scale-mixture representation of half-Cauchy distributions used to build the Gibbs updates."},{"cited_title":"A simple sampler for the horseshoe estimator,","cited_arxiv_id":null,"evidence_quote":"Provides the simple Gibbs sampler for the horseshoe estimator that the robust samplers extend."},{"cited_title":"Posterior inference in bayesian quantile regression with asymmetric laplace likelihood,","cited_arxiv_id":null,"evidence_quote":"Documents the need for a posterior variance adjustment under the Laplace working likelihood, the obstacle the paper claims the horseshoe prior overcomes in high dimensions."},{"cited_title":"Uncertainty quantification for the horseshoe (with discussion),","cited_arxiv_id":null,"evidence_quote":"Shows horseshoe credible intervals can be calibrated in sparse normal means models, motivating the extension to robust regression."},{"cited_title":"Is seeing believing? a prac- titioner’s perspective on high-dimensional statistical inference in cancer genomics stud- ies,","cited_arxiv_id":null,"evidence_quote":"Proposes the spike-and-slab robust Bayesian LASSO whose exact-sparsity inference results are the benchmark and whose simulation design is reused."},{"cited_title":"Horseshoe prior bayesian quantile regression,","cited_arxiv_id":null,"evidence_quote":"Proposes slice-sampling horseshoe Bayesian quantile regression, the main competing sampler the Gibbs approach is compared against."}],"review_version":1}