{"id":"5af56e1e-ae69-4de8-ad2b-70348aa23a50","arxiv_id":"2601.13514","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Thinning the score variable makes model selection and standard confidence intervals asymptotically independent, so ordinary inference is valid after penalized model selection.","lead":"This paper shows that a simple trick—adding and subtracting carefully chosen noise—lets researchers use ordinary statistical software for valid inference after model selection with penalized regression. It applies to generalized linear models and clustered data, sidestepping bespoke post-selection inference methods.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 6 hinges on a local-alternative assumption (Assumption 1(i)) that the simulations and application violate, so the empirical results do not actually exercise the theorem's regime.","rationale":"The reader identified Condition 4 as the weakest assumption, but my independent reading points to Assumption 1(i) (local alternatives) as the most load-bearing condition for the central claim. Assumption 1(i) is not a minor technicality: the proof of Theorem 6 explicitly uses it to guarantee that the selection-event set A^E_n converges to a set with non-empty interior and thus that the denominator in the conditional Berry-Esseen bound (Lemma 2) is bounded away from zero. Without this, the proof does not go through. The paper's simulations and application use fixed signals, so they do not satisfy this assumption. This is a gap between the theory and the empirical evidence, but it is not an internal inconsistency in the theorem itself. The theorem is correct as stated under Assumption 1; the problem is that the paper's practical claims are broader than the theorem. This supports a CONDITIONAL verdict: the main result is plausible but requires either (i) proving the result under fixed alternatives (possibly with a different proof technique) or (ii) explicitly repositioning the simulations as heuristic evidence and adding local-alternative simulations. My concern is partially aligned with the reader's rationale, which also notes the local-alternative mismatch, though the reader's formal weakest_assumption was Condition 4. I therefore set agreement_with_reader to 'partial'. The verdict remains UNCHANGED because my concern reinforces the CONDITIONAL evaluation rather than overturning it.","tokens_in":36284,"tokens_out":21490,"duration_ms":209160,"concrete_test":"Replicate the logistic regression simulation of Section 5.1 (Figure 2) under two data-generating schemes: (A) fixed signal as in the paper (θ* with ten nonzero entries, constant in n); (B) local alternative θ* = θ0/√n for a fixed sparse θ0. Run both for n = 200, 500, 1000 with the same λ scaling and γ = 1, and compute the conditional coverage of the 90% CIs from Algorithm 1 over the selected models. If coverage in (B) is close to 90% while coverage in (A) is also close (as the paper reports), then the local-alternative assumption is not restrictive in practice. If coverage in (B) drops below nominal, the theorem's proof does not cover the fixed-signal regime and the paper's simulations are not evidence for the central claim as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 6: conditional coverage ≥ 1−α for a standard CI after selection by the noisy penalized M-estimator. The proof of this theorem relies essentially on Assumption 1(i), which requires √n θ^{E,*}_n → θ^{E,*} (a fixed vector). This assumption is used (in the proof of Theorem 6, around Eq. B.47 and E3) to ensure that the limit set A^E has non-empty interior and that the denominator Pr(Z(1)_n ∈ (A^E_n)^{−ε}) is bounded away from zero, so the conditional-probability bound in Lemma 2 actually vanishes. If instead θ^{E,*}_n is fixed (not shrinking at n^{−1/2}), then √n θ^{E,*}_n diverges; the set A^E_n shifts to infinity, the Gaussian probability of any fixed ε-shrinkage can go to 0, and the bound in Lemma 2 becomes vacuous. Consequently, the asymptotic conditional-coverage guarantee is only established under local alternatives. Yet the simulations in Section 5.1 generate outcomes from a fixed sparse θ* (ten nonzero entries) without any n-dependent scaling, and the application uses real data with fixed effects. These settings violate Assumption 1(i) in the direction √n θ^{E,*}_n → ∞. The paper's discussion acknowledges some limitations (n>p, L1-only penalty) but does not flag this mismatch, so the reported coverage rates do not validate the theorem's assumptions. This is a load-bearing gap because it means the advertised 'no bespoke methods' result is proven only in a regime that the paper does not empirically test.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes a method for post-selection inference after penalized M-estimation. The main idea is to add Gaussian noise to the score of an M-estimator and use the two resulting noisy score variables for selection and inference, respectively. Under regularity conditions, the authors show that the selection event and the event that a standard fixed-model confidence interval covers its target can both be approximately characterized by these thinned score variables, yielding asymptotic conditional coverage (Theorem 6). A corollary applies the result to generalized linear models by adding noise directly to the outcomes (Corollary 7). The paper includes a Berry--Esseen-type analysis, simulation studies in linear, logistic, and cluster-correlated settings, and an application to a social network.","tokens_in":36609,"tokens_out":5831,"duration_ms":64253,"significance":"If the main theorem is correct, this is a substantial contribution: it would provide a simple, distribution-free, asymptotically valid alternative to bespoke selective inference procedures for a broad class of penalized M-estimators. The paper builds on and extends data-thinning ideas, and the proof machinery—especially the conditional Berry--Esseen bound—is nontrivial and largely carefully developed. The authors also provide code, which is valuable for reproducibility. However, as detailed below, the theorem is proved only under a local-alternative assumption that the paper's own simulations and application do not satisfy, and the proof has a gap concerning convergence of the variance estimator. The conceptual promise is real, but the current manuscript needs additional work before the central claim can be regarded as fully established.","major_comments":[{"comment":"Please note the above comment is the same as the intended comment; the section field is used for citation.","section":"Assumption 1(i) and Section 5.1"},{"comment":"This is a separate technical issue from the local-alternative mismatch.","section":"Appendix B.5 (proof of Theorem 6)"},{"comment":"This issue is local but affects the empirical implementation.","section":"Algorithm 1, Step 2"}],"minor_comments":[{"comment":"There are several typographical issues, e.g., \"independen\" in the abstract and formatting glitches like \"glmnetinR\". A careful proofread is recommended.","section":"Abstract and general presentation"},{"comment":"The condition \\ell_\\theta(Y_i) = \\theta^T A_i Y_i + a_{\\theta,i} is stated with A_i of dimension p x d. In the GLM application, Y_i is scalar and A_i = -X_i, so A_i \\bar W_{n,i} is p-dimensional. This is clear enough, but it would help to explicitly state the dimensions in Section 3.2 to avoid confusion.","section":"Section 3.2, Corollary 7"},{"comment":"The sentence \"We note the following.\" and the list formatting could be improved. Also, the claim that \"All other approaches attain 90% coverage\" is contradicted by the displayed undercoverage of Huang et al. in Figure 1; the text should be clarified.","section":"Section 5.1"},{"comment":"The discussion lists limitations n > p and L1-only penalty, but as noted above it does not mention the local-alternative restriction or the lack of validation in that regime. The discussion would be more transparent if this were included.","section":"Section 6"}],"recommendation":"major_revision","confidential_remarks":"The paper has a strong and interesting core idea, and the proof machinery is impressive. The main issues are: (1) the theorem's local-alternative assumption is not satisfied by the paper's own simulations and application, making the empirical claims not directly supportive of the theorem; (2) the proof of Theorem 6 assumes convergence of the variance estimator that is not part of the stated assumptions. Both are fixable, but they are load-bearing. I would recommend major revision with a request to address these points and to correct the Algorithm 1 typo."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline is this: the score-thinning construction is a genuine step forward in post-selection inference, and the central theorem is probably right under its stated assumptions—but the experiments are run in a regime the theorem doesn't cover, and the paper doesn't say so.\n\nWhat's new: rather than deriving a bespoke selective distribution for each penalty, the authors show that if you add carefully chosen Gaussian noise to the score (or, for GLMs, to the outcomes), the model selection event and the coverage event of a standard CI can both be written as events involving two approximately independent normal score variables. That lets you use ordinary software for the whole pipeline. The construction is clean, and it extends the thinning idea to logistic regression and cluster-correlated data, where previous data-thinning/fission results didn't apply. The proofs are detailed; Lemmas 4–5 and Theorem 6 have a real Berry-Esseen argument, and I didn't find a hole in that reasoning.\n\nWhere it's soft. The biggest issue is that Theorem 6 relies on Assumption 1(i), a local-alternative condition that keeps the selection event non-degenerate in the limit. The simulations and the network application use a fixed sparse truth, so sqrt(n) θ^{E,*}_n blows up. In that case the selection probability for any given wrong model goes to zero, the denominator in Lemma 2 isn't bounded away from zero, and the proof of Theorem 6 doesn't apply. The paper acknowledges the local-alternative assumption when it introduces it, but never flags that its own empirical section doesn't live in that regime. That's a real mismatch, and it should be addressed—either by a theory for fixed alternatives (or at least a heuristic argument) or by a clear statement that the simulations are a sanity check rather than a confirmation of the theorem.\n\nSecond, Condition 3 is assumed rather than proved in general. For GLMs they show it holds, but the theorem statement would be cleaner if they gave a general sufficient condition. Relatedly, the proof of Theorem 6 needs consistency of Ŝ^E_n, which isn't stated in Assumption 1. These are fixable.\n\nThe paper is honestly written, with limitations (n>p, L1-only) disclosed. I'd send it to a referee. It deserves one.\n\nAll best.","headline":"Score-thinning is a real advance, but the theory and simulations operate in different asymptotic regimes—still worth a serious referee.","tokens_in":37135,"tokens_out":5149,"would_cite":true,"duration_ms":48601,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62J07","62F12","62E20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Ordinary confidence intervals can be made valid after model selection by splitting a score variable with noise.","keywords":["post-selection inference","selective inference","penalized M-estimation","score variable","data splitting","randomized lasso","confidence intervals","asymptotic coverage"],"falsifier":"Simulate a setting where the Taylor remainder is not $o_p(n^{-1/2})$ (for example, median regression with fixed p and n) and measure the conditional coverage of the proposed interval; a persistent gap below $1-\\alpha$ would refute the theorem's reach. Alternatively, check whether Condition 4 holds for a given loss by verifying the remainder rate directly.","tokens_in":36108,"feed_emoji":"🎲","tokens_out":4422,"duration_ms":42090,"temperature":0.7,"texified_at":"2026-08-05T20:48:27.277933+00:00","pith_summary":"This paper claims that post-selection inference for penalized M-estimators does not require bespoke selective-inference pipelines. The key observation is that both the event that a penalized M-estimator selects a given model and the event that a standard confidence interval covers its target can be expressed as events involving one approximately normal score variable. If that score is split into two approximately independent pieces by adding and subtracting carefully scaled Gaussian noise, the selection event and the coverage event become asymptotically independent. As a result, standard confidence intervals centered on a noisy estimator attain the nominal coverage conditional on the selected model. The paper works out the theory for fixed p and growing n, and shows it holds for generalized linear models, where comparable thinning methods do not apply.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":9270,"prompt_tokens":718,"completion_tokens":8552,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":718,"completion_tokens_details":{"reasoning_tokens":7890}},"feed_headline":"Noise-split score makes post-selection inference standard","feed_subtitle":"Add and subtract noise from one score variable; ordinary confidence intervals then cover given the selected model.","key_machinery":"The score variable $\\hat{Z}_n = \\sqrt{n} P_n \\dot{\\ell}_{\\bar{\\theta}_{n}^{E,*}}$, the normalized gradient of the loss at the population parameter of the selected model. Gaussian thinning: with $W \\sim N(0, \\hat{\\Sigma})$, set $Z^{(1)} = \\hat{Z} + \\gamma W$ and $Z^{(2)} = \\hat{Z} - (1/\\gamma) W$; these are approximately independent normals. The selection event equals $\\{Z^{(1)} + o_p(1) \\text{ in a union of convex sets}\\}$ (Lemma 4), and the coverage event equals $\\{Z^{(2)} + o_p(1) \\text{ in a convex set}\\}$ (Lemma 5). Lemma 2's conditional Berry-Esseen bound converts this approximate independence into a conditional-coverage guarantee in Theorem 6.","core_discovery":"The central claim is Theorem 6: under regularity conditions, a standard (non-selective) confidence interval centered at the noisy model-E M-estimator covers the population parameter conditional on the event that the noisy penalized M-estimator selected model E, with asymptotic probability at least $1-\\alpha$. The proof decomposes the score variable into two approximately independent normal variables, shows the selection event depends only on one piece (up to a vanishing remainder) via the KKT conditions, and the coverage event only on the other, then applies a conditional Berry-Esseen bound. Corollary 7 extends the result to adding noise directly to the outcomes whenever the loss is affine in t","pith_inferences":["Editorial inference: the same score-splitting mechanism could plausibly be adapted to provide valid post-selection p-values and false-discovery-rate control, not just confidence intervals, for penalized M-estimators.","Editorial inference: Condition 5 restricts the approach to penalties whose selection regions are finitely piecewise-convex (true for the lasso but open for the group lasso); extending it would cover more structured penalties.","Editorial inference: a promising stress test is the p > n regime, which the paper leaves open; score thinning may behave differently when the score covariance is singular.","Editorial inference: the asymptotic independence between selection and inference suggests that the variance estimator and the noise split gamma could be tuned separately, hinting at a possible power-improving modification."],"forward_implications":["For practitioners, valid post-selection inference after lasso-style selection reduces to fitting two M-estimators, one with noise added and one with noise subtracted; no special selective-inference software is required.","The result covers generalized linear models with canonical links, including logistic regression, a setting where exact data-thinning methods are unavailable.","Coverage is conditional on the selected model, not merely unconditional, so the guarantee is the one selective-inference users actually want.","Intervals from this approach are wider than those from exact randomized selective inference in simulations, but narrower than sample splitting, offering a trade-off between convenience and power.","Corollary 7 implies that for appropriate losses, the analyst may add noise to the response instead of to the optimization objective, which is computationally simpler."],"fun_headline_variants":["Post-selection CIs via noise-split score thinning","Add noise to data for valid selection inference","Standard inference after selection? Just add noise","Score thinning enables ordinary confidence intervals"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The loss function's gradient must admit a uniform second-order Taylor expansion around the selected-model parameter whose remainder is $o_p(n^{-1/2})$; if that quick decay fails, both the event characterizations and the coverage guarantee collapse.","fun_headline_variants_meta":{"raw":{"variants":["Post-selection CIs via noise-split score thinning","Add noise to data for valid selection inference","Standard inference after selection? Just add noise","Score thinning enables ordinary confidence intervals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000587,"raw_usage":{"total_tokens":2617,"prompt_tokens":791,"completion_tokens":1826,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":535,"completion_tokens_details":{"reasoning_tokens":1770}},"tokens_in":535,"tokens_out":1826,"duration_ms":17927,"temperature":1.0,"reasoning_tokens":1770,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T09:30:43.070363+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate a setting where the Taylor remainder is not $o_p(n^{-1/2})$ (for example, median regression with fixed p and n) and measure the conditional coverage of the proposed interval; a persistent gap below $1-\\alpha$ would refute the theorem's reach. Alternatively, check whether Condition 4 holds for a given loss by verifying the remainder rate directly.","supporting_citations":[],"review_version":1}