{"id":"6ee77ee0-65c2-4545-8106-095c93c6d0cc","arxiv_id":"2606.25729","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Bootstrap calibration of generalized posterior credible sets is a level-specific scale correction, not a general shape fix, under fixed-dimensional asymptotics via higher-order Edgeworth expansions.","lead":"The paper uses Edgeworth expansions to analyze how bootstrap selects a learning rate to improve frequentist coverage of credible sets from generalized posteriors. It concludes that this approach corrects scale for specific levels but does not fix general shape problems unless covariances match proportionally.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly isolates the technical step (uniform coverage approximation plus local identification) required for root consistency; the abstract already flags the fixed-dimensional Edgeworth setting. Because the proportionality claim follows immediately once the Gaussian limit is taken, and no contradictory assumption is apparent, the UNVERDICTED verdict is left unchanged.","tokens_in":1652,"tokens_out":252,"duration_ms":14974,"concrete_test":"Re-derive the coverage probability up to o(1/sqrt(n)) in the Gaussian case (setting all Edgeworth correction terms to zero) and confirm that the equation for the calibrated learning rate becomes independent of nominal level alpha if and only if posterior covariance equals a scalar multiple of sampling covariance.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract derives the proportionality requirement directly from the leading Gaussian term in the coverage expansion after separating sampling and posterior Edgeworth corrections. The consistency statement for the bootstrap root is conditioned on the stated uniform-coverage and local-identification assumptions under fixed-dimensional regularity; these are explicit rather than hidden. No internal gap between the claimed Gaussian-limit behavior and the higher-order analysis is visible from the provided summary.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper develops a theory for bootstrap calibration of a scalar learning rate in generalized posterior credible sets to achieve frequentist coverage. Under regular fixed-dimensional asymptotics it derives higher-order coverage expansions via Edgeworth series, separating sampling corrections for the estimator from posterior corrections for boundaries, centers, and shapes; analyzes the stochastic approximation step in the calibration algorithm; establishes consistency of the bootstrap root for a fixed nominal level under a uniform coverage approximation and local identification; and shows that a single scalar learning rate can calibrate all nominal levels in the Gaussian limit only when posterior and sampling covariances are proportional. Hence bootstrap calibration supplies a level-specific scale correction rather than a remedy for general shape misspecification.","tokens_in":1725,"tokens_out":367,"duration_ms":36810,"significance":"If the derivations hold, the work supplies a precise characterization of when and why bootstrap calibration succeeds for generalized posteriors. The separation of the two Edgeworth sources, the explicit consistency result under stated assumptions, and the derivation of the proportionality requirement directly from the leading Gaussian term constitute substantive theoretical contributions. These results deliver falsifiable predictions about failure modes (non-proportional covariances) and clarify the method's scope, which is valuable for generalized Bayesian inference.","major_comments":[],"minor_comments":[{"comment":"The abstract refers to 'the implemented algorithm' without indicating its pseudocode or convergence criterion; a short description or reference to the relevant section would improve readability.","section":"Abstract"},{"comment":"Notation for the coverage function, learning rate, and the two Edgeworth correction terms should be introduced with a compact table or displayed equation early in the introduction to aid cross-referencing.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive assessment, the accurate summary of our results, and the recommendation for minor revision. We are pleased that the contributions are viewed as substantive.","responses":[],"tokens_in":1213,"tokens_out":53,"duration_ms":9034,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The central result is that a single learning rate calibrates coverage across nominal levels in the Gaussian limit only when posterior and sampling covariances are proportional; otherwise bootstrap calibration remains a level-by-level scale fix and does not correct general shape mismatch.\n\nThe separation of the two Edgeworth corrections is the clearest new piece. It makes explicit why the calibration step behaves the way it does and why the proportionality condition appears at the leading term. The consistency claim for the bootstrap root under uniform coverage and local identification is stated directly and rests on standard fixed-dimensional regularity, so the logic does not appear circular.\n\nThe main limitation is the maintained assumption of regular fixed-dimensional asymptotics. Generalized posteriors are often used precisely in settings where those conditions are questionable, yet the paper does not discuss how the proportionality requirement or the level-specific nature might change under weaker or high-dimensional regimes. The stochastic approximation analysis is mentioned but its practical convergence rate or sensitivity to tuning is not developed.\n\nThis is a focused theoretical note for readers who already work with Edgeworth expansions and generalized posteriors. A statistician interested in the frequentist calibration of robust or approximate Bayesian procedures will get a precise characterization that was not spelled out before. The argument is internally consistent on its own terms and the claims are stated with the necessary assumptions visible.\n\nI would send it to referees.","headline":"The paper cleanly separates sampling and posterior Edgeworth terms to show that scalar bootstrap calibration for generalized posteriors is level-specific unless the two covariances are proportional.","tokens_in":2195,"tokens_out":347,"would_cite":false,"duration_ms":9767,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A scalar learning rate calibrates generalized posterior credible sets for all nominal levels only when posterior and sampling covariances are proportional.","keywords":["generalized posterior","bootstrap calibration","credible sets","coverage probability","Edgeworth expansion","learning rate","shape misspecification","frequentist coverage"],"falsifier":"A simulation in which posterior and sampling covariances are not proportional yet a single scalar learning rate produces correct frequentist coverage for two or more distinct nominal levels would falsify the proportionality requirement.","tokens_in":2542,"feed_emoji":"","tokens_out":651,"duration_ms":14368,"temperature":0.7,"pith_summary":"The paper derives higher-order coverage expansions for generalized posteriors calibrated by bootstrap. It shows that the bootstrap coverage equation has a consistent root for a fixed nominal level under uniform approximation and local identification. The expansions separate sampling corrections for the estimator from posterior corrections for boundaries and shapes. In the Gaussian limit this implies that one scalar learning rate can adjust coverage across all levels only if the two covariances are proportional. Bootstrap calibration therefore acts as a level-specific scale adjustment rather than a general remedy for misspecification.","feed_headline":"Scalar learning rate calibrates posteriors only when covariances proportional","feed_subtitle":"Bootstrap adjustment corrects scale mismatch for one nominal level but cannot fix arbitrary shape differences across levels.","key_machinery":"The bootstrap coverage equation solved by stochastic approximation, together with the higher-order Edgeworth expansions that separate sampling and posterior contributions to coverage error.","core_discovery":"Using Edgeworth expansions under regular fixed-dimensional asymptotics, the root of the bootstrap coverage equation is consistent for any fixed nominal level. The expansions isolate two distinct error sources: the sampling Edgeworth term for the point estimator and the posterior Edgeworth term for credible-set location, scale, and shape. Consequently a single scalar learning rate calibrates all nominal levels in the Gaussian limit only when posterior covariance is proportional to sampling covariance, so bootstrap calibration remains a level-specific scale correction rather than a fix for arbitrary shape mismatch.","pith_inferences":["Practitioners could test proportionality of the two covariances before trusting a single calibrated learning rate across multiple levels.","The result suggests exploring vector or matrix learning rates when shape mismatch is detected.","The same decomposition may apply to other calibration methods that adjust a single parameter of the generalized posterior."],"forward_implications":["For any fixed nominal level the calibrated learning rate converges to the value that equates bootstrap and target coverage.","Coverage error decomposes additively into a sampling term and a posterior term, each expandable to higher order.","When covariances are proportional the same scalar works uniformly across nominal levels; otherwise each level requires its own scalar.","Shape misspecification between posterior and sampling distributions cannot be removed by any scalar adjustment."],"fun_headline_variants":["Only proportional covariances allow scalar posterior calibration","Bootstrap calibration is level-specific without covariance match","Edgeworth terms show bootstrap fixes scale not shape","Calibration consistent only when covariances proportional"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The root of the bootstrap coverage equation remains consistent under a uniform coverage approximation, local identification, and the regular fixed-dimensional asymptotics required for the Edgeworth expansions.","fun_headline_variants_meta":{"raw":{"variants":["Only proportional covariances allow scalar posterior calibration","Bootstrap calibration is level-specific without covariance match","Edgeworth terms show bootstrap fixes scale not shape","Calibration consistent only when covariances proportional"]},"model":"grok-4.3","cost_usd":0.005934,"raw_usage":{"total_tokens":2792,"prompt_tokens":622,"num_sources_used":0,"completion_tokens":53,"cost_in_usd_ticks":59337000,"prompt_tokens_details":{"text_tokens":622,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2117,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":622,"tokens_out":53,"duration_ms":12258,"temperature":1.0,"reasoning_tokens":2117,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-25T20:23:18.974636+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A simulation in which posterior and sampling covariances are not proportional yet a single scalar learning rate produces correct frequentist coverage for two or more distinct nominal levels would falsify the proportionality requirement.","supporting_citations":[],"review_version":1}