{"id":"697087d7-e49c-44ca-914d-bbbce82e6fbd","arxiv_id":"2606.00867","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Shapley-value anomaly tests equal simpler single-term tests for independent sensors but differ for correlated bivariate Gaussians, with strict superiority or inferiority depending on correlation sign.","lead":"The paper proves that Shapley-value anomaly tests for sensors match a simpler single-term test exactly when observations are independent, but produce different decision regions and error rates when sensors are correlated, with one sometimes strictly better depending on correlation sign. Practitioners choosing anomaly localization methods in sensor networks may want to account for these conditions before defaulting to Shapley values.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the premise that enables all subsequent derivations. With the full manuscript now available, the proofs remain conditional on exactly those modeling choices; no further load-bearing gap appears.","tokens_in":1859,"tokens_out":252,"duration_ms":28645,"concrete_test":"Re-derive the decision regions and error probabilities for the bivariate Gaussian case (constant attack, large positive correlation) directly from the value function v(S) without invoking the Shapley formula; confirm that the single-term test and full-Shapley test produce the stated error-probability ordering.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims consist of explicit equivalence and strict comparison results derived for independent observations and for specific bivariate Gaussian/Laplacian distributions with additive attacks. These rest on the existence of optimal binary classifiers for every coalition (as stated) and on the concrete value-function definition used to compute the Shapley values. The abstract and described proofs contain no visible internal inconsistency, circularity, or unsupported step once those premises are granted; the reported superiority/inferiority depending on correlation sign is a direct consequence of the differing decision regions under dependence.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript analyzes the use of Shapley values for localizing sensor anomalies/attacks when optimal binary classifiers are employed in the value function. It proves equivalence (identical error probability) between a full Shapley-value test and a single-term test for independent observations. For two-sensor dependent cases (correlated bivariate Gaussian/Laplacian densities with constant or Gaussian additive attacks), the tests produce distinct decision regions and error rates; the Shapley test is shown to be strictly superior or inferior depending on correlation sign and magnitude, with a suggestion that combining the approaches yields a strictly better detector. Numerical illustrations are included.","tokens_in":1964,"tokens_out":412,"duration_ms":17171,"significance":"If the derivations hold, the work supplies the first explicit statistical comparison of Shapley-based anomaly localization against simpler alternatives, establishing when the two coincide exactly and when one dominates. The parameter-free proofs on concrete distributions, together with the explicit superiority/inferiority results that depend on correlation sign, constitute a clear, falsifiable contribution that can guide practical use of Shapley values in sensor security.","major_comments":[],"minor_comments":[{"comment":"The abstract states that proofs are given for the dependent bivariate cases, but the main text should include an explicit statement of the attack magnitude range over which the strict superiority/inferiority holds (e.g., a lemma or corollary after the decision-region derivation).","section":"§4"},{"comment":"Notation for the value function v(S) and the single-term test should be introduced with a short table or equation block early in §2 to avoid repeated parenthetical definitions later.","section":"§2"},{"comment":"Figure captions for the numerical results should state the exact correlation values, attack type, and SNR used in each panel so that the plots can be reproduced without consulting the text.","section":"§5"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The summary accurately captures the core contributions regarding equivalence for independent sensors and the strict superiority/inferiority results for correlated bivariate cases.","responses":[],"tokens_in":1311,"tokens_out":62,"duration_ms":10871,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that an optimized Shapley-based test for localizing anomalies is exactly equivalent to a lower-complexity test that uses only one term from the Shapley calculation when the sensor observations are independent. For two-sensor correlated Gaussian or Laplacian cases with additive attacks, the tests produce different decision regions and error rates, and which one is better flips with the sign of the correlation.\n\nThe paper supplies the first explicit proofs of these equivalence and ordering results. It works with mathematically optimal binary classifiers on the listed distributions and derives the error probabilities directly. That is new relative to the empirical Shapley applications it cites, and the approach avoids circularity by starting from the probability models rather than fitting parameters.\n\nThe assumptions are stated up front: optimal classifiers exist for every coalition, and the observations follow the specific independent or bivariate distributions with constant or Gaussian attacks. Those premises are reasonable for a theoretical analysis, though they do limit how far the ordering results extend. No internal contradictions appear in the claims.\n\nThis is targeted work for researchers in statistical signal processing who already use or evaluate Shapley values for sensor anomaly tasks. A reader outside that niche will not find much to take away. The proofs are narrow but cleanly executed, so the paper deserves a serious referee to verify the derivations in the full text rather than a desk reject.","headline":"They prove Shapley anomaly tests match a simpler single-term test for independent sensors but can be strictly better or worse than it in correlated bivariate Gaussian cases depending on correlation sign.","tokens_in":2455,"tokens_out":347,"would_cite":false,"duration_ms":12472,"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":"For independent sensor observations the Shapley value anomaly test is exactly equivalent to a simpler single-term test with identical error probability, while the two differ and one can outperform the other in correlated bivariate cases dep","keywords":["Shapley value","sensor anomaly localization","statistical analysis","independent observations","correlated bivariate Gaussian","anomaly detection","optimal binary classifiers"],"falsifier":"Explicit computation of the error probability for a bivariate Gaussian model with large positive correlation and additive anomaly, showing whether the Shapley-value test or the single-term test achieves the lower error rate.","tokens_in":2751,"feed_emoji":"","tokens_out":802,"duration_ms":18607,"temperature":0.7,"pith_summary":"The paper proves that when sensor observations are independent an optimized Shapley-value anomaly test reduces to a lower-complexity test that uses only one term from the Shapley calculation and produces the same probability of error. In contrast, for statistically dependent observations drawn from correlated bivariate Gaussian or Laplacian distributions with constant or Gaussian attacks the two tests generate different decision regions and different error probabilities. The analysis further shows that the full Shapley test is sometimes strictly worse and sometimes strictly better than the single-term test, with the outcome governed by the sign of the correlation when its magnitude is large. The results indicate that the two approaches can be combined to obtain a strictly superior test in the dependent setting.","feed_headline":"Shapley anomaly test equals single-term version for independent sensors","feed_subtitle":"In correlated bivariate cases the two tests differ, with relative performance set by correlation sign; combining them improves results.","key_machinery":"Comparison of the full Shapley value versus a single term from its calculation inside optimal binary classifiers for deciding whether a given sensor observation is anomalous.","core_discovery":"We prove that for cases with independent sensor observations, an optimized anomaly test using the Shapley value is equivalent to an optimized lower-complexity anomaly test using a single term in the Shapley value calculation, yielding the exact same probability of error. For some popular dependent observation cases involving two sensors, including correlated bivariate Gaussian/Laplacian probability density functions and constant/Gaussian attacks/anomalies, we prove that these two tests are fundamentally different, yielding different decision regions and error probabilities. Further, we prove that the Shapley value test is sometimes strictly inferior to the other test in certain statistically","pith_inferences":["The equivalence result may extend to other joint distributions that satisfy conditional independence, allowing the simpler test to be used more broadly.","In networks with many sensors the computational saving from the single-term test could matter for real-time localization even when observations are only approximately independent.","A practical system could first test for independence among sensors and then select or combine the two methods accordingly."],"forward_implications":["For independent observations the lower-complexity single-term test can be used in place of the Shapley-value test without any increase in error probability.","In the examined dependent bivariate cases the two tests produce different decision regions, so their error probabilities are not the same.","When correlation is large the relative performance of the two tests reverses with the sign of the correlation coefficient.","A test that combines the Shapley value and the single term can achieve strictly lower error probability than either alone in the dependent bivariate setting."],"fun_headline_variants":["Shapley test equals single-term for independent sensors","Shapley differs from single-term in bivariate dependent cases","Correlation sign sets relative performance of two anomaly tests","Shapley strictly inferior in some dependent Gaussian scenarios","Combined approach beats both in certain correlated cases"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The existence of mathematically defined optimum binary classifiers together with the assumption that sensor observations follow either independence or the specific correlated bivariate Gaussian and Laplacian distributions with the listed attack types.","fun_headline_variants_meta":{"raw":{"variants":["Shapley test equals single-term for independent sensors","Shapley differs from single-term in bivariate dependent cases","Correlation sign sets relative performance of two anomaly tests","Shapley strictly inferior in some dependent Gaussian scenarios","Combined approach beats both in certain correlated cases"]},"model":"grok-4.3","cost_usd":0.006307,"raw_usage":{"total_tokens":3014,"prompt_tokens":767,"num_sources_used":0,"completion_tokens":72,"cost_in_usd_ticks":63074500,"prompt_tokens_details":{"text_tokens":767,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2175,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":767,"tokens_out":72,"duration_ms":15824,"temperature":1.0,"reasoning_tokens":2175,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T17:50:43.499752+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Explicit computation of the error probability for a bivariate Gaussian model with large positive correlation and additive anomaly, showing whether the Shapley-value test or the single-term test achieves the lower error rate.","supporting_citations":[],"review_version":1}