{"id":"f3d3aeee-dd95-45f9-9dc5-bdacfbfcbc3e","arxiv_id":"2607.06939","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":4,"one_line_summary":"The compensator-based inference framework for signal detection under unknown background is extended from i.i.d. data to binned Poisson counts, with asymptotic guarantees and a Fermi LAT case study.","lead":"This paper extends a statistical method for detecting signals (like dark matter) against an unknown background from unbinned to binned Poisson count data. It matters because binned data is the standard format in modern physics and astronomy experiments, making the method practical for real-world searches.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The Riemann sum error bound in Eq. (19) of Appendix A.2 does not converge to zero as stated; the proof of Proposition 1 has a gap for d≥2 and a loose bound even for d=1.","rationale":"The reader correctly identified the appendix proofs as the weakest link but focused on the regularity conditions (A1)-(A6) rather than the specific gap in the Riemann sum bound. The concern I've identified is more concrete: equation (19) contains a bound that provably does not converge to zero as written, creating a gap in the proof of Proposition 1. For d=1, this gap is benign—the actual quantity does vanish, just faster than the bound suggests. For d≥2, the gap may be substantive: the condition required by Lemma 1 may genuinely fail under C^1 smoothness.\n\nSince the paper claims d-dimensional generality in the introduction and throughout, but all examples and case studies are d=1, and the proof gap affects the stated scope, I recommend CONDITIONAL rather than ACCEPT. The fix is straightforward: either restrict claims to d=1 (matching all examples) or strengthen the smoothness assumption to C^2 (which would give Riemann sum error O(k^{-2/d}), making √T·O(k^{-2/d}) → 0 for d<4). The core contribution for d=1 binned data is sound and valuable, and the Fermi LAT case studies are unaffected. Code and data are publicly available, which is a positive signal. The concern is about proof correctness for the stated level of generality, not about the fundamental methodology.","tokens_in":22462,"tokens_out":8786,"duration_ms":431401,"concrete_test":"Re-derive the bound in Eq. (19) explicitly for d=2 with a specific C^1 function S_0 (e.g., S_0(x,y) = sin(x)cos(y) on [0,1]²) and uniform k bins with T=ck. Numerically compute √T · |∑ f_i S_0(x_i) - ∫ S_0 dF| for increasing k. If the quantity does not converge to zero, Lemma 1 cannot be applied and Proposition 1 fails for d≥2. If it does converge (e.g., due to midpoint-rule cancellation), the bound in (19) still needs correction but the result may hold.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the proof of Proposition 1 (Appendix A.2, equation 19), the authors verify the condition √T ∑ f_i ξ_i → 0 from Lemma 1 by bounding the Riemann sum approximation error. The stated bound is:\n\n‖𝒳‖_d · sup_{x∈𝒳} max_i ‖∂_i S_0(x)‖ · √(Td/k) → 0\n\nHowever, under the asymptotic regime T/k → c ∈ (0,∞), the factor √(Td/k) → √(dc), a nonzero constant. The bound therefore converges to ‖𝒳‖_d · sup · √(dc) ≠ 0, not to zero as claimed.\n\nThe issue is that the bound is missing the bin diameter factor. For a d-dimensional grid with k bins of equal volume v = Vol(𝒳)/k, the bin diameter scales as O(k^{-1/d}). The Riemann sum error for a C^1 function is O(k^{-1/d}), so √T · O(k^{-1/d}) = O(√c · k^{1/2-1/d}), which → 0 only for d=1. For d=2 it converges to a nonzero constant; for d≥3 it diverges.\n\nThis means Lemma 1's conditions may not be verifiable for d≥2 under the assumed C^1 smoothness, and Propositions 1-3 may not hold in the stated d-dimensional generality. For d=1 (all case studies), the conclusion is correct—the Riemann sum error is O(1/k) and √T·O(1/k) → 0—but the proof's bound is too loose by a factor of √k. The same gap propagates to equations (39) in the proof of Proposition 2 and (48) in the proof of Proposition 3, which rely on analogous Riemann sum approximations.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"This manuscript extends the compensator-based inference framework of Banerjee and Algeri (2026) for signal detection under an unknown background to the setting of binned Poisson count data. The asymptotic regime considered is T/k → c ∈ (0,∞), where both the total expected counts T and the number of bins k grow, keeping the counts Poisson rather than Gaussian. The authors derive asymptotically Gaussian estimators for the signal intensity η when a labeled background-only sample is available (Propositions 1–2) and conservative inference when it is not (Proposition 3). The framework is illustrated on Fermi LAT simulated data with varying binning resolutions and proposal background distributions.","tokens_in":23037,"tokens_out":1397,"duration_ms":322932,"significance":"The extension to binned Poisson data is well-motivated by common practice in physics and astronomy. The key practical contribution is that the compensator δ absorbs the misspecification between the postulated and true background, so inference on η does not require estimating the unknown background density f_b. The case study in Section 3.1 demonstrates robustness across four proposal backgrounds and three binning resolutions, which is a compelling empirical validation. The provision of reproducible R code and data (GitHub link in the manuscript) is a strength. The proofs in Appendices A.1–A.6 are detailed and follow standard asymptotic arguments; Lemma 1 correctly handles the Poisson regime via moment generating function calculations.","major_comments":[{"comment":"Appendix A.2, Eq. (19): The Riemann sum error bound used to verify the condition √T ∑ f_i ξ_i → 0 from Lemma 1 appears to have a gap. The stated bound is ‖𝒳‖_d · sup_{x∈𝒳} max_i ‖∂_i S_0(x)‖ · √(Td/k), which under the regime T/k → c converges to ‖𝒳‖_d · sup · √(dc), a nonzero constant, not zero as claimed. The bound is missing the bin diameter factor, which scales as O(k^{-1/d}). For d=1, the corrected bound gives √T · O(1/k) → 0, so the conclusion holds for all case studies. For d≥2, the argument as written does not establish the condition under C^1 smoothness. The authors should either restrict the main results to d=1 (which covers all examples) or strengthen the smoothness assumptions (e.g., C^2 with a tighter Riemann sum bound) to cover d≥2. This issue propagates to the analogous bounds in Eqs. (39) and (48).","section":null},{"comment":"Section 4, Eq. (13) and surrounding text: The choice of the dominating term in g_β involves free parameters σ_0, ε, μ_1, μ_2, and λ. The manuscript states that σ_0 should be 'substantially larger than the width of the true signal' and that μ_1, μ_2 should be 'sufficiently close to the boundaries of M_ε,' but does not provide guidance on how to select these in practice when f_b is unknown. The sensitivity analysis (right panel of Figure 1) partially addresses this for λ, but the dependence on the other parameters is not explored. A brief discussion of robustness to these choices, or at minimum a recommendation for default selection rules, would strengthen the practical applicability of Proposition 3.","section":null}],"minor_comments":[{"comment":"Abstract: The text has lost all spaces between words (e.g., 'Theproblemofsignaldetectionunderanunknownbackgroundcanbeframedasoneofinferring'). This appears to be a formatting artifact and should be corrected.","section":null},{"comment":"Section 2, Eq. (5): The orthonormal expansion of f_b/g includes δ S†(x), but the relationship between δ and the L²(G) projection of f_b/g onto S† could be stated more explicitly for readers unfamiliar with the framework.","section":null},{"comment":"Table 1: The p-values for k=100 are reported as 8.761×10⁻⁷ and 8.870×10⁻⁷ (three significant figures) while k=30 and k=50 entries use four significant figures. Minor inconsistency in precision.","section":null},{"comment":"Appendix A.3: The regularity conditions (A1)–(A6) are stated but not thoroughly motivated. A brief remark connecting each condition to its role in the proofs (e.g., (A1) for concavity of the log-likelihood, (A5) for interiority of β*) would improve readability.","section":null},{"comment":"Section 3.1: The signal is described as 'a Gaussian bump, centered at 3.5 GeV with standard deviation 0.35 GeV,' but in the log-transformed specification f_s(x) ∝ exp{−(exp(x)−3.5)²/(2·0.35²)}·exp(x). The Jacobian term exp(x) is correctly included, but a sentence clarifying that this is the density of the log-transformed variable would help.","section":null},{"comment":"References: The citation 'Banerjee and Algeri (2026)' appears throughout but the arXiv identifier 2605.20508 has a 2026-style prefix. If this is a forthcoming publication, the reference should be updated to the published version if available.","section":null},{"comment":"Figure 1, right panel: The legend entries for different λ values are somewhat difficult to distinguish. Consider using a colorblind-friendly palette or adding line type variations beyond what is currently shown.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The skeptic's concern about the Riemann sum bound in Eq. (19) is technically correct and is the most substantive issue. For d=1, the result holds and all case studies are one-dimensional, so the practical impact is limited. However, the manuscript's abstract and Section 1 claim d-dimensional generality, which is not supported by the proof as written. The authors can resolve this by either restricting to d=1 or adding a C² assumption with a tighter bound. This is a fixable issue that does not undermine the core contribution. The paper is a solid extension of the compensator framework and merits publication after this clarification."},"author_rebuttal":null,"desk_editor":{"model":"glm-5.2","letter":"The main thing to know: this paper extends the compensator framework from Banerjee and Algeri (2026) to binned Poisson counts under the regime T/k → c, which is the right asymptotic model for modern physics experiments. That extension is genuine and useful. Lemma 1 cleanly handles the Poisson regime, and Propositions 1–3 give asymptotically Gaussian estimators for the signal intensity with and without a background-only sample. The Fermi LAT case studies show the method is robust to the choice of postulated background, which is the practical selling point. Code and data are on GitHub. This is a real contribution to the subfield, not a reshuffling of prior work.","headline":"Solid extension of compensator-based inference to binned Poisson data; one real proof gap for d≥2 that needs fixing but doesn't touch the d=1 case studies.","tokens_in":23330,"tokens_out":2553,"would_cite":false,"duration_ms":90639,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Background-free signal detection extends to binned Poisson data","keywords":["signal detection","unknown background","model misspecification","binned data","Poisson counts","compensator","mixture models","conservative inference"],"falsifier":"If one constructs a postulated background density g_beta for which log g_beta is not concave in beta, or a signal density f_s that is not continuously differentiable, and shows that the test statistics T1, T2, or T3 fail to converge to standard normal — either through systematic bias or incorrect variance — the framework's applicability would be materially narrowed.","tokens_in":22724,"feed_emoji":"📊","tokens_out":1306,"duration_ms":154182,"temperature":0.7,"pith_summary":"The paper extends a previously established inferential framework — compensator-based inference — from independent identically distributed data to the setting of binned Poisson counts, the data format most common in modern physics and astronomy experiments. The core problem is detecting a known signal density mixed on top of an unknown background density. Rather than estimating the background density directly (which propagates uncontrolled uncertainty), the approach replaces it with a postulated density and introduces a single parameter, called the compensator (delta), that absorbs the discrepancy between the postulated and true background. When a labeled background-only sample is available, the compensator can be estimated directly, yielding asymptotically Gaussian estimators of the signal intensity eta whose validity does not depend on how close the postulated background is to the truth. When no background-only sample exists, the framework provides conservative but valid inference by testing a lower bound on eta whose conservativeness is guaranteed whenever the compensator is non-positive — a condition that can be assessed via a sensitivity analysis on the postulated background. The paper proves all three results (Propositions 1-3) under the asymptotic regime where total counts and number of bins grow together (T/k converges to a constant), ensuring the Poisson character of the data is preserved rather than approximated by a Gaussian limit.","feed_headline":"Detect signals without knowing the background, even in binned data","feed_subtitle":"A one-parameter 'compensator' absorbs background misspecification, extending valid signal detection to Poisson-count histograms common in物理学","key_machinery":"1) The compensator identity: eta = (theta - delta) / (||S||_G - delta), which isolates background misspecification into a single scalar delta. 2) Lemma 1, which establishes asymptotic normality of weighted Poisson count sums under the regime T/k -> c, serving as the engine for all three propositions. 3) The sensitivity analysis construction (equation 13) for the no-background-sample case, which injects a diffused dominating Gaussian component into the postulated background to guarantee non-positivity of the compensator and thus conservative inference.","core_discovery":"The central mechanism is that the signal intensity eta can be decomposed as (theta - delta) / (||S||_G - delta), where theta is the expectation of the normalized score function under the data distribution, delta is the same expectation under the background-only distribution (the compensator), and ||S||_G is a known norm of the score function under the postulated background. This decomposition means that the entire effect of background misspecification is captured by one scalar parameter. In the binned Poisson regime, theta and delta can each be estimated as simple weighted sums of bin counts multiplied by the score function evaluated at bin centers. Their asymptotic normality follows from a泊","pith_inferences":["The compensator framework could potentially extend to multidimensional or correlated binning schemes (e.g., spatial-spectral joint bins), since the key identity is distribution-free and the Poisson structure generalizes naturally, though the regularity conditions would need re-verification.","The sensitivity analysis for the no-background-sample case relies on the signal being a localized bump; for extended or multi-modal signals, the condition ensuring non-positivity of the compensator may not hold, limiting the conservative-inference route to bump-hunting scenarios.","If the signal density f_s itself carries nuisance parameters (the paper mentions this as possible but does not develop it), the compensator identity would need modification, and the score function S would depend on those parameters, potentially complicating the clean decomposition.","The framework implicitly assumes the signal and background are additive in the density mixture; non-additive contamination models (e.g., convolution-based or selection-effect-dominated scenarios) may not admit the same compensator decomposition."],"forward_implications":["Physics experiments that bin data into histograms for signal searches can apply valid statistical inference without requiring accurate background models, as long as a background-only control sample is available for estimating the compensator.","When no background-only sample exists, the sensitivity analysis provides a principled way to trade detection power for guaranteed conservatism — the researcher can visually inspect how much 'padding' in the postulated background is needed to ensure valid conservative inference.","The framework's robustness to the choice of postulated background (demonstrated empirically with uniform, exponential, and Gaussian-tail proposals all yielding similar p-values) suggests that practitioners need not invest heavily in background modeling, only in obtaining a background-only sample.","The asymptotic regime T/k -> c covers the practically important case where bin counts remain small (Poisson, not Gaussian), which is common in high-energy physics and X-ray astronomy where fine binning is used to preserve spectral resolution."],"fun_headline_variants":["One parameter absorbs background misspecification in binned signal detection","Binned Poisson counts: valid signal detection without knowing the background","Compensator method extends to binned histogram signal detection","Signal detection in binned data survives an unknown background","A single scalar captures all background error in binned Poisson inference"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The regularity conditions (A1)-(A6) — particularly the concavity of the log-postulated-density in its parameters and the continuous differentiability of the known signal density — are assumed sufficient for asymptotic normality in the binned Poisson regime. If these conditions fail for commonly used background model families, the Gaussian approximations underlying all three test statistics may not hold.","fun_headline_variants_meta":{"raw":{"variants":["One parameter absorbs background misspecification in binned signal detection","Binned Poisson counts: valid signal detection without knowing the background","Compensator method extends to binned histogram signal detection","Signal detection in binned data survives an unknown background","A single scalar captures all background error in binned Poisson inference","Binned signal detection without estimating the misspecified background","Unknown background in binned counts? One compensator keeps inference valid","From iid to binned Poisson: compensator-based signal detection extends"]},"model":"glm-5.2","effort":"low","cost_usd":0.0,"raw_usage":{"total_tokens":1136,"prompt_tokens":462,"completion_tokens":674,"prompt_tokens_details":null},"tokens_in":462,"tokens_out":674,"duration_ms":21793,"temperature":1.0,"reasoning_tokens":611,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T22:31:07.648364+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If one constructs a postulated background density g_beta for which log g_beta is not concave in beta, or a signal density f_s that is not continuously differentiable, and shows that the test statistics T1, T2, or T3 fail to converge to standard normal — either through systematic bias or incorrect variance — the framework's applicability would be materially narrowed.","supporting_citations":[],"review_version":1}