{"id":"a676f438-f46c-4ec8-a698-d1a8e2e1466c","arxiv_id":"1908.09112","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A disjunct support spike-and-slab prior makes Bayes factors grow exponentially with sample size in quasi-sparse regression, unlike full-support or point-mass-spike priors.","lead":"This paper proposes a new Bayesian prior for regression that ignores tiny nonzero coefficients and only keeps effects larger than a user-chosen threshold. The authors prove that the resulting model comparison scores favor the correct sparse model with growing data, where standard spike-and-slab priors fail.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1 is asserted with a convergence statement but proved only via a lower bound; the missing upper bound leaves the central Bayes-factor consistency claim unproven as stated.","rationale":"Reading in good faith: the paper proposes a clean interval-support spike-and-slab prior and identifies a real failure mode of full-support priors under quasi-sparseness. The simulations are consistent with the claimed behavior, and the lower-bound argument is a reasonable skeleton for a proof. The single most load-bearing weakness is that the headline theorem does not establish what it says: 'converges to c' requires both liminf and limsup, and the proof supplies only liminf via an upper bound on the denominator that discards a polynomial factor. The boundary of the truncated-normal priors is the place where a naive Laplace argument could fail, so the missing upper bound is a concrete correctness risk, not just exposition. The same proof gap interacts with scope limitations: d is treated as fixed (the (d+1)/2 log n term and the O_p(1) determinant), while the high-dimensional experiments use d=1000; and the adaptive δ procedure used in the experiments is not covered by the theorem. These issues are addressable, consistent with the reader's CONDITIONAL verdict; I do not see evidence that the central idea is wrong, so I do not move the verdict. Agreement with the reader is partial: the reader names the adaptive-δ scope as the weakest assumption, whereas I regard the missing upper bound in Theorem 1 as more load-bearing, with the scope limitations as contributing factors.","tokens_in":43850,"tokens_out":20407,"duration_ms":226093,"concrete_test":"Complete the proof analytically: for S' that swaps one variable (e.g., S'=(S\\{j})∪{k}), derive an asymptotic lower bound on the denominator marginal likelihood by applying Laplace's method on the closure of Θ_S' and integrating over a shrinking neighborhood of the constrained MLE θ̂_{S',n}, including the boundary value |β_k|=δ. If the expansion gives log p(y|X,S') = n E_x[g(θ_S')] - (d+1)/2 log n + O_p(1), then (1/n) log BF → c_∆, validating Theorem 1 as stated. If the boundary produces a different polynomial order (e.g., n^{-1/2} instead of n^{-1} in one coordinate), verify whether (1/n) log BF still converges to c_∆; the theorem statement, but not necessarily the exponential-growth claim, would need revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central theoretical result, Theorem 1, states that (1/n) log p(y_n|X_n,S)/p(y_n|X_n,S') converges to c>0 for every S'≠S. The proof, however, derives only a lower bound. After the Laplace expansion for the numerator, Eq. (3), it bounds the denominator by p(y|X,S') ≤ p(y|X, θ̂_{S'}) (Eq. 4), i.e., by the maximized likelihood, discarding the polynomial n^{-(d+1)/2} factor. This yields log BF ≥ n c_∆ - (d+1)/2 log n + O_p(1), hence liminf of (1/n) log BF ≥ c_∆, but nothing about limsup. To prove convergence to a constant, one needs a matching lower bound on the denominator, presumably via a Laplace expansion over the closed support; the spike/slab boundary points |β_j|=δ are excluded from Θ_S' and can alter the polynomial order, so this is not a purely cosmetic step. The proof also treats d as fixed: the term (d+1)/2 log n is O(1) only for fixed d, and the Laplace determinant is O_p(1) only then. The high-dimensional experiments (d=1000) and the data-driven δ rule of Section 6 are therefore outside the theorem's scope. The lower bound alone does establish exponential growth from below, so the core idea is plausible, but the theorem as stated is overclaimed.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a new spike-and-slab prior for linear regression in which the spike and slab have disjoint supports: the spike is a truncated normal on [-δ,δ] and the slab is a scale mixture of truncated normals on the complement. The threshold δ is intended to encode practical relevance, so that coefficients of magnitude smaller than δ are treated as negligible even if nonzero (quasi-sparseness). The central theoretical claim is Theorem 1, which states that for the true model S and any other model S', (1/n) log p(y_n|X_n,S)/p(y_n|X_n,S') converges in probability to a positive constant c, i.e., Bayes factors grow exponentially. Theorem 2 claims that full-support spike-and-slab priors give Bayes factors that are bounded in probability, and Theorem 3 claims that product moment matching priors are unsuitable in the quasi-sparse setting because they eventually select all nonzero coefficients. The paper also gives a Gibbs sampler for posterior model probabilities, a data-driven rule for choosing δ based on estimated MSE increase, and extensive simulations and real-data comparisons.","tokens_in":44185,"tokens_out":9907,"duration_ms":107351,"significance":"If Theorem 1 were fully established as stated, the paper would offer a practical prior class that combines a meaningful practical-significance threshold with provably growing Bayes factors, filling a real gap for quasi-sparse regression. The strengths are a clean prior construction, analytically tractable conditional updates for the Gibbs sampler, and a thorough empirical comparison including high-dimensional settings and several real data sets. Theorems 2 and 3 are useful and mostly straightforward. However, the central theorem is only proved via a lower bound, and the proof's assumptions exclude the high-dimensional and data-driven-δ settings used in the main experiments. These issues make the theoretical contribution weaker than advertised, though the qualitative exponential-growth claim appears plausible and potentially fixable.","major_comments":[{"comment":"The theorem asserts convergence of (1/n) log BF to a positive constant, but the proof only establishes a lower bound. After the Laplace expansion for the numerator, the denominator is bounded above by the maximized likelihood p(y|X, θhat_{S'}), discarding the integration measure. This gives log BF ≥ n c_Δ - (d+1)/2 log n + O_p(1), hence only liminf (1/n) log BF ≥ c_Δ > 0. No matching upper bound for the Bayes factor (equivalently, no lower bound for the wrong-model marginal likelihood) is supplied. Because the wrong-model support is closed and has boundaries at ±δ, a Laplace-type lower bound is not a cosmetic step: boundary contributions can affect the polynomial-order term. The theorem should either be restated as a liminf/at-least-exponential-growth result, which is sufficient for the paper's qualitative claim, or the missing upper bound should be proved.","section":"Section 4, Theorem 1 and proof after Eq. (3)-(4)"},{"comment":"The proof of Theorem 1 treats the dimension d as fixed: the factor (d+1)/2 log n and the Laplace determinant O_p(1) depend on that. The high-dimensional experiments in Section 7 use d=1000 with n=100 and n=1000, which are outside the theorem's scope; nonetheless Section 7.1 states that these experiments confirm Theorem 1. Additionally, Theorem 1 assumes a fixed δ chosen before data and |β_j,t| ≠ δ, whereas the main 'proposed*' method in Section 6 selects δ from the data via the estimated MSE increase. The paper should explicitly state these scope limitations and either prove an extension for growing d and adaptive δ or soften the claims that the theorem covers the reported experiments.","section":"Section 4, Theorem 1 assumptions vs. Section 7"},{"comment":"The quantities reported as 'Bayes factors' are not clearly the marginal-likelihood ratios used in Theorem 1. The MCMC procedure in Section 5 estimates posterior model probabilities p(S|y,X); converting these to Bayes factors requires multiplying by the prior odds p(S')/p(S), and the paper does not describe such a conversion. Moreover, the alternative model S' is chosen as the most frequently selected model from the same data, so the comparison is data-dependent rather than a fixed-model Bayes factor. The '∞' entries for cases where the true model is always selected are also not finite Bayes factors and prevent the stated averaging. The authors should clarify the estimand or report Bayes factors against a fixed alternative such as the null or full model.","section":"Section 7.1 and Table 1"}],"minor_comments":[{"comment":"The notation 'P→ Op(1)' is nonstandard; the statement should say that the Bayes factor is bounded in probability, i.e., O_p(1).","section":"Theorem 2 statement"},{"comment":"In the displayed chain near the end of the proof, the notation 'P→' is used on expressions that still contain n; this is informal and should be rewritten as a statement about (1/n) log BF or with explicit o_p(n) error terms.","section":"Section 4, proof of Theorem 1"},{"comment":"Equation (5) reads 'ι(R,σ1²) ∝ ι(R,σ1²) = (2πσ1²)^{1/2}', which is a tautology in the first part; it should state that ι(R,σ1²) ≈ (2πσ1²)^{1/2} for large σ1² and small δ, as the text intends.","section":"Section 5.4, Eq. (5)"},{"comment":"There are typos in the tables: 'Gracia-Donato' should be 'Garcia-Donato', and 'CUNFUC' in Table 29 should be 'CONFUC'. Appendix D also contains 'Tn Table 27' instead of 'In Table 27'.","section":"Tables 6, 7, and 29"},{"comment":"The abstract and Section 7.1 say the experiments 'confirm' Theorem 1 in high-dimensional settings, but Theorem 1 does not apply there; this phrasing should be moderated to say the experiments are suggestive but outside the theorem's assumptions.","section":"Abstract and Section 7.1"}],"recommendation":"major_revision","confidential_remarks":"The reader's and skeptic's main concern is correct and lands on the manuscript: Theorem 1's proof establishes only a liminf, and the theorem statement overclaims convergence to a constant. The high-dimensional and adaptive-δ scope issues are also real. The paper is not beyond repair: the qualitative exponential-growth claim follows from the lower bound, and the Gibbs sampler plus empirical comparisons are valuable. However, the experimental 'Bayes factors' in Table 1 need to be redefined or recomputed as actual marginal-likelihood ratios against a fixed alternative. I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the central theorem is overclaimed. The proof of Theorem 1 derives a lower bound on (1/n) log BF of the form c - O(log n/n) via Laplace for the numerator and an upper bound for the denominator by maximizing the likelihood. That gives liminf >= c, not convergence to c. The missing matching upper bound is not cosmetic: the denominator's Laplace order depends on the boundary of the disjunct support at +/-delta, and the text excludes those points to keep Theta_S' open. So the theorem as stated is unproven.\n\nThat said, the core proposal is good and I think the qualitative conclusion is right. Replacing the point-mass spike at zero with an interval spike over [-delta,delta] is a natural way to handle quasi-sparseness, and the paper improves on Johnson-Rossell in exactly the quasi-sparse regime. The Gibbs sampler is carefully derived with analytic conditionals; the slice-sampler acceptance scheme is honest. Simulations show the expected behavior: Bayes factors grow with n, and the method keeps false positives under control where BIC/SSLASSO start selecting noise. Real data comparisons are thorough.\n\nSoft spots beyond the theorem: the proof treats d as fixed and the Laplace determinant is O_p(1) only then, while the main experiments go to d=1000. The data-driven delta rule from Section 6 is used in the asterisked results but has no asymptotic guarantee and is outside the theorem's assumptions. These are limitations, not fatal; they can be fixed by either proving a matching upper bound or restating the theorem as a liminf result, and by either extending the asymptotics to growing d or clearly reporting the fixed-d theorem as motivation rather than full coverage.\n\nThe citation pattern looks standard -- Chipman, Johnson-Rossell, Rockova-George, horseshoe literature all cited. No sign of missing key references.\n\nWho should read it: anyone doing Bayesian variable selection in quasi-sparse settings, and people working on Bayes factor inconsistency of full-support priors. I'd send it to a referee; it deserves a serious look, though I expect the proof needs to be substantially corrected. If the authors can state and prove the liminf version or close the gap, this is a publishable contribution.","headline":"The interval spike prior is a sound idea and the paper is worth engaging, but Theorem 1 proves only exponential growth from below, not the convergence to a constant the text claims.","tokens_in":44641,"tokens_out":1820,"would_cite":true,"duration_ms":19599,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F15","62J05","62F03"],"pacs":[],"model":"deepseek-v4-flash","headline":"Spike-and-slab priors with disjoint supports make Bayes factors grow exponentially, restoring consistent variable selection in quasi-sparse regression.","keywords":["quasi-sparse regression","spike-and-slab prior","Bayes factor consistency","disjunct support prior","variable selection","truncated normal prior","Gibbs sampling","practical significance threshold"],"falsifier":"Simulate a regression where a true coefficient is exactly $\\beta_j=\\delta$ (or drifts to $\\delta$ as $n$ grows) and check whether the log-Bayes-factor ratio in Theorem 1 still diverges to a positive $c$; alternatively, run the Section 6 adaptive $\\delta$ procedure on a quasi-sparse design with many coefficients just below $\\delta$ and test whether false positives accumulate at large $n$.","tokens_in":43668,"feed_emoji":"📈","tokens_out":7702,"duration_ms":65582,"temperature":0.7,"pith_summary":"The paper establishes that a spike-and-slab prior whose spike and slab have genuinely disjoint supports—the spike confined to the interval $[-\\delta,\\delta]$ and the slab to its complement—restores consistency to Bayesian variable selection when many true regression coefficients are non-zero but negligibly small (the quasi-sparse setting). Standard full-support spike-and-slab priors give Bayes factors between rival models that remain bounded as the sample size grows, so they cannot separate a practically relevant signal from negligible noise. Moment-matching nonlocal priors are consistent but place their spike only at the exact value $0$, so they treat every tiny non-zero coefficient as relevant and select overly large models under quasi-sparseness. The paper proves that, with the disjoint-support prior, the log Bayes factor favoring the true model grows linearly in $n$ (Theorem 1), and its experiments show fast-growing Bayes factors and better false-positive control than hard-thresholding alternatives.","feed_headline":"Split prior at δ makes Bayes factors grow exponentially","feed_subtitle":"In quasi-sparse regression, separating small coefficients from signals restores consistent model selection.","key_machinery":"The central object is the disjunct support spike-and-slab prior: a truncated normal spike $\\mathcal{N}_{[-\\delta,\\delta]}(0,\\sigma_0^2)$ and a truncated slab (a normal with an inverse-chi-square prior on its variance, giving a Cauchy-like tail) restricted to $|\\beta|\\ge\\delta$, with $\\sigma_0$ tuned so the two densities agree at $\\pm\\delta$. This single construction does two jobs: the threshold $\\delta$ encodes an interpretable definition of practical relevance, and the disjoint supports guarantee that any wrong subset assigns zero prior mass to the true coefficients, which is what drives the exponentially growing Bayes factors in Theorem 1; a Gibbs sampler over a latent inclusion vector $z$ then estimates the posterior model probabilities despite the non-conjugate priors.","core_discovery":"Formally, let $S=\\{j:|\\beta_{j,t}|>\\delta\\}$ be the set of practically relevant variables and $S'$ any other set. The paper proves in Theorem 1 that for its disjunct support priors, $(1/n)\\log\\,[p(y_n|X_n,S)/p(y_n|X_n,S')]\\xrightarrow{P} c$ for some $c>0$, so the Bayes factor grows exponentially fast in the sample size. The reason is that the true coefficient vector of a relevant variable lies outside the support of every wrong model's prior; the wrong model's marginal likelihood is then bounded by its constrained maximum likelihood, while the true model's marginal likelihood follows the Laplace approximation, leaving a positive gap in expected log-likelihood. Two companion results sharpen the contrast: Theorem 2 shows that full-support spike-and-slab priors (as in Chipman et al., 2001) yield Bayes factors that are only $O_p(1)$, and Theorem 3 shows that product moment-matching priors (Johnson and Rossell, 2012) make the Bayes factor of the relevant set against the full model converge to $0$ in probability in the quasi-sparse setting, precisely because their spike sits at $\\{0\\}$ rather than on an interval.","pith_inferences":["The same disjoint-support construction could plausibly repair Bayes-factor inconsistency in other model classes with quasi-sparse structure, such as generalized linear models or Gaussian graphical models, since the proof only relies on the true parameter falling outside the wrong model's support.","Because Theorem 1 requires $\\delta$ fixed in advance, the adaptive $\\delta$ selection of Section 6 is an open theoretical gap; a fully hierarchical prior on $\\delta$ might let the data determine practical relevance while preserving consistency.","The exponential growth rate $c$ suggests a natural sequential-testing rule: stop collecting data once the log Bayes factor crosses a pre-specified threshold, which could give sample-size savings in screening applications."],"forward_implications":["Bayes factors under the proposed prior increase at rate $e^{nc}$, so Bayesian hypothesis tests can separate practically relevant covariates from negligible noise once the sample is large enough.","Full-support spike-and-slab priors give bounded Bayes factors, so their use for formal model testing is questionable even though they are common in practice.","Product moment-matching priors, while consistent for exact zeros, will asymptotically prefer the model containing every non-zero coefficient, which is the wrong answer in the quasi-sparse setting.","The latent-variable Gibbs sampler makes the non-conjugate model computationally usable, and the threshold $\\delta$ provides an interpretable control on the sparsity-versus-prediction trade-off."],"supporting_citations":[{"why":"Defines the full-support spike-and-slab prior that the paper shows yields only $O_p(1)$ Bayes factors (Theorem 2).","marker":"Chipman et al. (2001)"},{"why":"Introduces product moment-matching priors with disjunct support at $\\{0\\}$, the object of Theorem 3 and the main alternative in the quasi-sparse comparison.","marker":"Johnson and Rossell (2012)"},{"why":"Establishes the consistent-Bayes-factor theory for nonlocal priors whose support excludes $\\{0\\}$, the template the paper extends to a $\\delta$-interval spike.","marker":"Johnson and Rossell (2010)"},{"why":"Provides the Laplace approximation that the proof of Theorem 1 uses for the marginal likelihood of the true model.","marker":"Kass et al. (1990)"},{"why":"Supplies the classical result on hypothesis testing with disjunct supports that motivates the asymptotic argument.","marker":"Walker (1969)"},{"why":"Justifies the Beta-Binomial prior on model size that gives the method its multiplicity control.","marker":"Scott and Berger (2006)"}],"fun_headline_variants":["Disjunct spike-and-slab priors give exponential Bayes factors","Split support at delta yields consistent Bayes factor growth","Disjunct priors: Bayes factors grow exponentially in n","Separate tiny effects: consistent Bayes factors for quasi-sparse","New prior gives exponential Bayes factor growth in quasi-sparse"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of Theorem 1 assumes the true coefficients never fall exactly on the boundary $\\pm\\delta$ and that $\\delta$ is fixed before the data are collected; the paper's recommended adaptive choice of $\\delta$ by expected MSE increase (Section 6) carries no such asymptotic guarantee.","fun_headline_variants_meta":{"raw":{"variants":["Disjunct spike-and-slab priors give exponential Bayes factors","Split support at delta yields consistent Bayes factor growth","Disjunct priors: Bayes factors grow exponentially in n","Separate tiny effects: consistent Bayes factors for quasi-sparse","New prior gives exponential Bayes factor growth in quasi-sparse"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000955,"raw_usage":{"total_tokens":4130,"prompt_tokens":1062,"completion_tokens":3068,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":678,"completion_tokens_details":{"reasoning_tokens":2987}},"tokens_in":678,"tokens_out":3068,"duration_ms":21461,"temperature":1.0,"reasoning_tokens":2987,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:22:07.483220+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate a regression where a true coefficient is exactly $\\beta_j=\\delta$ (or drifts to $\\delta$ as $n$ grows) and check whether the log-Bayes-factor ratio in Theorem 1 still diverges to a positive $c$; alternatively, run the Section 6 adaptive $\\delta$ procedure on a quasi-sparse design with many coefficients just below $\\delta$ and test whether false positives accumulate at large $n$.","supporting_citations":[],"review_version":1}