{"id":"fb2022fb-e7e1-4c41-b118-0985d76a432c","arxiv_id":"2502.02674","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Constraint-aware confidence intervals for ill-posed inverse problems are made computationally practical via a Berger-Boos bounding set, sampling, and quantile regression, achieving nominal coverage with shorter intervals than OSB.","lead":"A new method computes valid confidence intervals for hard-to-measure quantities in constrained inverse problems by sampling a data-adaptive bounding region and calibrating test statistics. In high-energy physics unfolding simulations it reaches the target coverage with intervals up to 19 percent shorter than the standard OSB method.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Practical coverage depends on the sampler and quantile regressor not missing high-quantile regions; Section 5.1.2 concedes the Polytope sampler often misses polytope corners, so 'achieves nominal coverage both theoretically and in practice' is not supported for the implemented intervals.","rationale":"The paper's central theoretical contribution—the Berger–Boos transfer (Lemma 3.1) and the oracle Global/Sliced intervals—is sound; the union-bound argument in Appendix A is correct. The soft spot is the bridge from oracle cutoffs to the implemented sampling-plus-quantile-regression cutoffs. The reader identified this as the weakest assumption, and I agree. I sharpen the concern: the quantity being estimated is a supremum of a quantile surface over Bη, while the proposed tools (uniform Polytope samples plus gradient boosting) are designed for integration and regression, not for finding suprema. Lemma B.1's convergence of the empirical max requires the sampler to hit every open set; the paper's own admission about corners violates this in finite time. The quantile-regression consistency is assumed, not established, for gradient boosting in an 80-dimensional problem. Hence the implemented intervals should be described as asymptotically calibrated under unverified assumptions, not as achieving nominal coverage 'both theoretically and in practice.' This does not change the verdict: CONDITIONAL is right because the oracle theory is correct and the simulations are encouraging, but the paper should either add a conservative correction to the cutoff (e.g., a margin informed by sampler/regressor error) or provide a diagnostic that the supremum is captured. The Lemma B.2 misstatement is a secondary internal flaw in the Sliced Optimized proof, reinforcing the need for revision.","tokens_in":32917,"tokens_out":14464,"duration_ms":135397,"concrete_test":"Re-run the Section 6.3 coverage study replacing Algorithm 2's gradient-boosting predictions at test points with direct Monte Carlo quantile estimates (Algorithm 1) with N=10^4 at the same Polytope-sampled design points; additionally evaluate Q_x(1−γ) by direct MC at the SSB-extreme points x_l, x_u from Equation (35) and at the Chebyshev center, and compare to the fitted bq_γ(x). If the direct-MC empirical maximum exceeds bq at any of those points, the implemented cutoff is biased low and the reported coverage is unreliable; if coverage using direct MC quantiles is still at least nominal, the quantile regression is the weak link; if neither changes coverage, the concern is rebutted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The oracle coverage guarantee (Lemma 3.1) is correct for the idealized cutoffs q_{γ,η}(μ) and q_{γ,η}. The implemented intervals replace these cutoffs with (i) bq = max_k q̂_γ(x_k) from Algorithm 2 and (ii) bm_γ(μ) from Section 3.2. Those estimates preserve coverage only if they do not fall below the true suprema over Bη. Theorem 4.1 assumes away the two failure modes: uniform consistency of the gradient-boosting quantile regressor is merely assumed, and Lemma B.1 requires the sampler to have positive mass on every open ball of Bη. Section 5.1.2 states that the Polytope sampler 'in practice ... often has some difficulty reaching the corners of the generated polytope'; those corners are exactly where the LLR quantile surface of a constrained problem tends to be largest (cf. Section 6.2, where a dedicated importance-like sampler was needed to reach the high-quantile boundary). A downward-biased bq lowers the LLR acceptance cutoff, shortens the interval, and produces under-coverage. Thus the abstract's 'both theoretically and in practice' claim is not warranted by the theorems: the practical claim rests only on the specific simulations. In addition, the proof of Sliced Optimized convergence (Theorem 4.1 item 4, Appendix B.4) uses Lemma B.2 whose key condition is misstated: for ε>0 it requires |f(μ)| > δ' iff |μ−μ*| < ε, the opposite of the separation condition needed, and it cannot hold for the relevant f. This weakens the theoretical support for one of the four headline variants.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes four confidence-interval constructions for a scalar functional φ(x*) in constrained, possibly rank-deficient inverse problems with additive Gaussian noise and known parameter constraints. The construction restricts the test-inversion intervals of Batlle et al. (2023) to a data-adaptive Berger–Boos set Bη, defines sliced and global maximum quantiles q_{γ,η}(µ) and q_{γ,η}, and proves an oracle coverage guarantee in Lemma 3.1 and Corollary 3.2. To implement the oracle cutoffs, the paper proposes two samplers (the VGS sampler and a Vaidya-walk Polytope sampler) and a quantile-regression estimator based on gradient boosting, and states a consistency theorem (Theorem 4.1) for the resulting Global/Sliced × Inverted/Optimized intervals. Numerical experiments in two- and three-dimensional constrained Gaussian settings and in an 80-dimensional wide-bin unfolding problem compare the intervals with the OSB interval.","tokens_in":33247,"tokens_out":9034,"duration_ms":81965,"significance":"If the implementation matched the oracle construction, the paper would be a useful step: it gives a principled way to reduce the conservatism of OSB intervals while retaining a frequentist coverage guarantee, and it demonstrates the approach on a realistic high-energy-physics unfolding problem. The oracle coverage lemma is a clean union-bound argument and the Berger–Boos budget is handled correctly in that lemma. However, the practical interval constructions rely on unproven and partly unmet conditions: uniform consistency of the quantile regressor, positive-mass sampling of Bη, and a correct stability lemma for the Sliced Optimized endpoint. The numerical evidence is encouraging but does not by itself establish the theorem-level claim for the implemented intervals, and no code is provided. The contribution is real, but the gap between the oracle theory and the implemented procedure needs to be closed or explicitly acknowledged.","major_comments":[{"comment":"The condition stated in Lemma B.2, namely that for all ε>0 there exists δ′>0 such that |f(µ)|>δ′ if and only if |µ−µ*|<ε, cannot hold for the functions to which the lemma is applied. At µ=µ* one typically has f(µ*)=0, so the left-hand side is false while the right-hand side is true; conversely, points outside the ε-neighborhood can have large |f|. The intended separation condition is presumably something like “|µ−µ*|≥ε implies |f(µ)|≥δ′”, and the proof of convergence of inf_{f_k≥0} to inf_{f≥0} requires that condition together with uniform convergence. As written, Lemma B.2 does not prove Statement 4 of Theorem 4.1.","section":"Appendix B.4 (Lemma B.2)"},{"comment":"The theorem assumes that the quantile regression in Algorithm 2 is consistent for all x, i.e., P(|bq_γ(x)−Q_x(1−γ)|>ε)→0 for every fixed x, but the implemented gradient-boosting estimator of Section 5.2 is not shown to satisfy this, and no regularity conditions or rates are given. In the proof, this assumption is used to control the first term of Eq. (S.7) for the empirical maximum over design points; pointwise consistency is not enough to control a maximum over M points as M grows, so a uniform consistency statement would be needed. Since the numerical experiments use Algorithm 2 with the gradient-boosting quantile regressor, the theoretical guarantee for the implemented intervals is not established.","section":"Section 4, Theorem 4.1"},{"comment":"Theorem 4.1 and Lemma B.1 require the design-point distribution to have positive mass on every Lebesgue-positive subset of Bη. The Polytope sampler of Algorithm 4 generates MCMC draws over a bounding polytope P_d ⊇ Bη, not i.i.d. draws from such a measure on Bη, and Section 5.1.2 states that in practice it “often has some difficulty reaching the corners of the generated polytope.” Those corners are exactly the regions where the LLR quantile of a constrained problem can be largest, and the paper’s own Section 6.2 needed a separate importance-like sampler (Algorithm 5) to reach the high-quantile boundary. If the sampler misses high-quantile regions, bq (or bm_γ) is biased downward, the LLR acceptance cutoff is too small, and coverage is lost. Thus the convergence theorem does not apply to the implemented sampler, and the abstract’s “both theoretically and in practice” claim is not supported for the implemented intervals.","section":"Section 5.1.2 vs. Lemma B.1 / Theorem 4.1"},{"comment":"The paper states that all four intervals “achieve nominal coverage … both theoretically and in practice.” The theoretical coverage in Lemma 3.1 concerns oracle cutoffs q_{γ,η}(µ) and q_{γ,η}; Theorem 4.1 provides only convergence in probability of the sampled endpoints to those oracle endpoints. That combination gives at best asymptotic coverage under the theorem’s assumptions, not a finite-sample guarantee, and the practical statement is based on selected simulations with 10^3 replications and no code. The wording should be softened to distinguish oracle coverage, asymptotic consistency of the implemented endpoints, and empirical performance.","section":"Abstract and Section 4"}],"minor_comments":[{"comment":"The remark says the Berger–Boos construction is “equivalent to a data-dependant reduction of the constraint set, replacing X for Bη both in the test statistic and the quantile optimization problems,” but the intervals in (23)–(24) use λ(µ,y;X) as defined in (11), with X, not Bη. Please clarify or correct the statement.","section":"Section 3.1, Remark 1"},{"comment":"In the definition of A2, Q_x(1−γ) should be Q_{x*}(1−γ) with µ*=φ(x*); as written, x is undefined.","section":"Appendix A.1, Lemma 3.1 proof"},{"comment":"The notation λ({(1−γ)N}) is unclear; it should be written as the order statistic λ_{(⌈(1−γ)N⌉)} or with an equivalent explicit expression.","section":"Algorithm 1, line 6"},{"comment":"The condition that there exists a point µ ∈ φ(Bη) satisfying λ(µ,y;X) < q_{γ,η} is stated to be equivalent to the interval being non-empty for linear models, but no proof of this equivalence is provided.","section":"Section 4, Theorem 4.1"},{"comment":"The paper reports that in the smooth setting the Global intervals and OSB dramatically over-cover, and in the adversarial setting the Sliced Optimized interval over-covers; these results are presented as strengths, but they also show that not all four constructions achieve close-to-nominal coverage in practice. The text should discuss why over-coverage is consistent with the stated goal that all four intervals achieve nominal coverage.","section":"Section 6.3"},{"comment":"No code or detailed hyperparameter settings for the gradient-boosting quantile regressor are provided beyond a reference to scikit-learn; given the tuning described in Section 5.2, this makes replication difficult.","section":"Section 5.2 and Section 6"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is built directly on Batlle et al. (2023), and the self-citations are appropriate. My main concern for the editor is that the abstract and conclusion overstate the support for the implemented intervals: Theorem 4.1 assumes rather than proves quantile-regression consistency, and the sampler admission in Section 5.1.2 undermines the “in practice” claim. The paper would be acceptable in principle after the theoretical statements are corrected and the claims are calibrated to what is actually proved. The absence of code is also a hindrance for a methods paper with heavy computation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Core of this paper is a correct and useful extension of Batlle et al. The Berger-Boos union-bound argument (Lemma 3.1) is clean and genuinely transfers the 1994 construction to constrained inverse problems. The four interval variants are a natural taxonomy, and the numerical work is serious: the 80-D unfolding study with 11-19% shorter intervals than OSB is a meaningful practical result. The authors are also honest about the samplers' limitations, e.g., Section 5.1.2 admits the Polytope sampler often has difficulty reaching corners.\n\nThe soft spots are real but don't sink the paper. First, the appendix has a misstated separation condition in Lemma B.2 — the 'if and only if' goes the wrong direction for the convergence of the Sliced Optimized interval. That leaves the proof of Theorem 4.1 item 4 with a gap. The other three convergence statements don't rely on this lemma, so the framework still stands. Second, the abstract says the intervals achieve nominal coverage 'both theoretically and in practice.' The theory covers the oracle intervals; the implemented intervals inherit coverage only if the sampler and quantile regressor don't miss high-quantile regions, which is assumed rather than proven. The simulations are consistent with coverage, but that's empirical evidence, not a theorem. Third, no code or data is provided, which makes the computational claims harder to verify.\n\nOverall, this is a solid methodological contribution with a correct central lemma and substantial numerical evidence. The Sliced Optimized proof needs fixing, and the abstract should be toned down to say 'asymptotically, under consistency of the sampler and quantile regression.' I'd send it out for review — a good referee can help the authors close the gap and sharpen the claims.","headline":"Correct Berger-Boos coverage lemma with a useful HEP unfolding demonstration, but the abstract overclaims practical coverage and one appendix lemma is misstated.","tokens_in":33840,"tokens_out":2980,"would_cite":true,"duration_ms":27518,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F25","62G08","65J22"],"pacs":[],"model":"deepseek-v4-flash","headline":"By shrinking the parameter search to a data-dependent Berger–Boos set and splitting the coverage budget, this paper constructs four confidence intervals that keep the stated coverage and shorten intervals in high-dimensional inverse…","keywords":["inverse problems","confidence intervals","test inversion","quantile regression","Berger–Boos set","ill-posed problems","unfolding","uncertainty quantification"],"falsifier":"For a fixed observation in any of the paper's low-dimensional examples, compute the true maximum quantile $q_{\\gamma,\\eta}$ by dense grid optimization over $B_\\eta$, then compare it with the sampled or regressed estimate from Algorithms 1 and 2; finding any observation where the estimate lies below the true maximum, or where a coverage simulation using an exhaustive sampler exceeds one using the Polytope sampler, would refute the practical claim.","tokens_in":32643,"feed_emoji":"📏","tokens_out":9303,"duration_ms":77545,"temperature":0.7,"pith_summary":"This paper tries to make the test-inversion confidence intervals of Batlle et al. (2023) practical and less conservative for functionals in ill-posed inverse problems. The key move is to replace the full constraint set by a data-dependent Berger–Boos set, a compact subset that contains the true parameter with probability at least $1-\\eta$, and then maximize the $1-\\gamma$ quantile of the log-likelihood-ratio statistic only over this subset, with $\\gamma$ chosen so that the total error budget $\\alpha$ is split as $\\gamma + \\eta$. The paper proves that this budget split preserves $1-\\alpha$ coverage and that four sampling-based interval constructions converge in probability to the oracle intervals. In an 80-dimensional particle-unfolding simulation, the Sliced variants reach nominal coverage and are 11 to 19 percent shorter on average than the standard one-at-a-time strict bounds interval. The payoff is a recipe for constraint-aware intervals that remain valid when the forward model is rank-deficient and the parameter space is high-dimensional.","feed_headline":"Data-shrunk intervals cut unfolding uncertainty by up to 19%","feed_subtitle":"A Berger–Boos set plus quantile regression yields valid 68% intervals in an 80-dimensional inverse problem.","key_machinery":"The load-bearing object is the Berger–Boos set $B_\\eta = \\{x \\in X : \\|y - f(x)\\|_2^2 \\le \\chi^2_{n,\\eta}\\}$, a $1-\\eta$ confidence set for $x^*$ obtained as the pre-image of a chi-squared ball under the forward model. The argument splits the coverage budget as $1-\\alpha = (1-\\eta)(1-\\gamma)$, so maximizing the larger $1-\\gamma$ quantile over $B_\\eta$ costs the same total error as maximizing the $1-\\alpha$ quantile over $X$. The LLR statistic $\\lambda(\\mu, y) = \\inf_{x \\in \\Phi_\\mu \\cap X} \\|y - f(x)\\|^2 - \\inf_{x \\in X} \\|y - f(x)\\|^2$ defines the tests being inverted; sampling design points inside $B_\\eta$, estimating the quantile surface by Monte Carlo percentiles or quantile regression, and then either inverting the test or using the estimated cutoff in an endpoint optimization yields the four interval constructions.","core_discovery":"The central claim is that valid confidence intervals for a one-dimensional functional can be built by inverting the constrained log-likelihood-ratio test over a shrunk, data-adaptive set instead of the entire constraint set. Specifically, Lemma 3.1 states that for any $\\eta \\in (0, \\alpha)$, the set $C_{\\mathrm{sl}}^{\\alpha}(y; B_\\eta) = \\{\\mu : \\lambda(\\mu, y) \\le q_{\\gamma,\\eta}(\\mu)\\}$ is a $1-\\alpha$ confidence set for $\\mu^* = \\varphi(x^*)$ whenever $\\gamma \\le \\alpha - \\eta$, where $q_{\\gamma,\\eta}(\\mu) = \\sup_{x \\in B_\\eta \\cap \\Phi_\\mu} Q_x(1-\\gamma)$; Corollary 3.2 gives the same guarantee for the global version. The paper further claims that the oracle quantiles can be approximated consistently: Theorem 4.1 shows that both inversion-based and optimization-based versions converge in probability to the oracle interval as the number of design points and LLR samples grows, assuming a consistent quantile regressor. Empirically, all four constructions reach nominal coverage in the tested settings, while the OSB interval fails in the 3D constrained-Gaussian case and in an adversarial 80D unfolding setting; the Sliced intervals reduce average length relative to OSB by 11 to 19 percent where OSB is valid.","pith_inferences":["The same $\\gamma + \\eta$ budget argument could be applied to any test statistic with a computable quantile, not only the LLR, as long as a $1-\\eta$ confidence set for $x^*$ is available; the paper notes the generalization in principle but does not pursue it.","The divergent behaviors of Sliced Inverted and Sliced Optimized in the simulations suggest the Optimized variant is more robust to local quantile-regression error, since it smooths the max-quantile curve; a single poorly predicted quantile can collapse the Inverted interval to a point.","One testable extension is to replace the Polytope sampler with a sampler biased toward high-quantile regions, as the appendix's importance-like sampler does for the 3D example; such bias could improve coverage robustness in higher dimensions without changing the budget argument."],"forward_implications":["If Lemma 3.1 and Corollary 3.2 hold, any $1-\\eta$ confidence set for $x^*$ can be plugged into the construction, so the framework extends beyond Gaussian pre-image ellipsoids to other confidence sets with the same budget split.","The convergence result in Theorem 4.1 means that, with enough samples and a consistent quantile estimator, the inverted and optimized intervals coincide with the oracle Berger–Boos interval, so users can choose either form without changing the asymptotic guarantee.","In the smooth 80-dimensional unfolding setting, the Sliced constructions cut expected interval length by 11.1 to 18.7 percent relative to OSB while keeping at least nominal coverage, implying that the added computational cost buys shorter intervals in realistic high-energy-physics problems.","In the adversarial unfolding setting, all four constructions maintain nominal coverage where OSB under-covers, so the method promises valid uncertainty quantification when the true parameter is concentrated near constraint boundaries."],"supporting_citations":[{"why":"Supplies the test-inversion framework, the LLR statistic, and the max-quantile definitions that this paper adapts to the Berger–Boos set.","marker":"Batlle et al. (2023)"},{"why":"Introduces the confidence-set-of-nuisance-parameter construction and the error-budget split that underlies Lemma 3.1.","marker":"Berger and Boos (1994)"},{"why":"Provides the Vaidya-walk MCMC algorithm used to sample the bounding polytope around the Berger–Boos set in high dimensions.","marker":"Chen et al. (2018)"},{"why":"Provides the p-ball sampling technique used by the VGS sampler for low-dimensional ellipsoidal Berger–Boos sets.","marker":"Voelker et al. (2017)"},{"why":"Defines the OSB interval and the wide-bin unfolding setup used as the main baseline and application.","marker":"Stanley et al. (2022)"},{"why":"Establishes the use of quantile regression on simulated data that Algorithm 2 adapts to the composite-null setting.","marker":"Dalmasso et al. (2020, 2024)"}],"fun_headline_variants":["Less conservative intervals for inverse problems","Cut uncertainty 19% in unfolding with data-adaptive intervals","Sampling trick tightens confidence intervals in inverse problems","Calibration method yields sharper interval bounds","Designs intervals that actually cover in hard cases"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The implemented intervals keep their coverage guarantee only if the sampler and the quantile regressor do not underestimate the maximum $1-\\gamma$ quantile over the Berger–Boos set; missing a high-quantile region makes the cutoff too small and the interval under-cover.","fun_headline_variants_meta":{"raw":{"variants":["Less conservative intervals for inverse problems","Cut uncertainty 19% in unfolding with data-adaptive intervals","Sampling trick tightens confidence intervals in inverse problems","Calibration method yields sharper interval bounds","Designs intervals that actually cover in hard cases"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000358,"raw_usage":{"total_tokens":2033,"prompt_tokens":1136,"completion_tokens":897,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":752,"completion_tokens_details":{"reasoning_tokens":827}},"tokens_in":752,"tokens_out":897,"duration_ms":9516,"temperature":1.0,"reasoning_tokens":827,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T11:30:32.742397+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed observation in any of the paper's low-dimensional examples, compute the true maximum quantile $q_{\\gamma,\\eta}$ by dense grid optimization over $B_\\eta$, then compare it with the sampled or regressed estimate from Algorithms 1 and 2; finding any observation where the estimate lies below the true maximum, or where a coverage simulation using an exhaustive sampler exceeds one using the Polytope sampler, would refute the practical claim.","supporting_citations":[{"cited_title":"Optimization-based frequentist confidence intervals for functionals in constrained inverse problems: Resolving the Burrus conjecture","cited_arxiv_id":"2310.02461","evidence_quote":"Supplies the test-inversion framework, the LLR statistic, and the max-quantile definitions that this paper adapts to the Berger–Boos set."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the confidence-set-of-nuisance-parameter construction and the error-budget split that underlies Lemma 3.1."},{"cited_title":"J., and Yu, B","cited_arxiv_id":null,"evidence_quote":"Provides the Vaidya-walk MCMC algorithm used to sample the bounding polytope around the Berger–Boos set in high dimensions."},{"cited_title":"R., Gosmann, J., and Stewart, T","cited_arxiv_id":null,"evidence_quote":"Provides the p-ball sampling technique used by the VGS sampler for low-dimensional ellipsoidal Berger–Boos sets."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the OSB interval and the wide-bin unfolding setup used as the main baseline and application."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the use of quantile regression on simulated data that Algorithm 2 adapts to the composite-null setting."}],"review_version":1}