{"id":"41211e19-9262-4334-900a-0e39be4c4719","arxiv_id":"2507.21406","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"A 'Shapley uncertainty' metric for LLM outputs is proposed, but its total equals the differential entropy it was meant to fix, and the claimed properties and performance gains are not supported.","lead":"This paper proposes a new uncertainty score for AI answers that uses semantic similarities between generated sentences instead of a simple right-or-wrong grouping. The authors claim it predicts AI mistakes better than existing methods, but the math behind the score contains a serious internal contradiction and the experimental gains are mixed.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Shapley efficiency makes the total Shapley uncertainty exactly the Gaussian differential entropy the paper rejects in Prop. 3.5; the two propositions cannot both hold, and the proof's key inequality is unverified.","rationale":"The paper's abstract and Section 3 present two connected contributions: three axiomatic properties and a proof that Shapley uncertainty satisfies them, plus an empirical claim that the metric predicts LLM correctness better than baselines. The theoretical contribution is the load-bearing part because it motivates the metric and distinguishes it from semantic entropy and plain differential entropy. The Shapley-efficiency identity Σ_i φ_i = v(N) - v(∅) is a standard theorem; with v(∅) = 0 it makes the Shapley total equal to h(s-tilde). Therefore the paper's own Prop. 3.5, which says h(s-tilde) violates Properties 3.2 and 3.3, is incompatible with Prop. 3.6, which says the Shapley sum satisfies them. The proof in Appendix A does not resolve this; it asserts the needed monotonicity inequalities without proving them, and those inequalities are precisely the crux. The reader's weakest_assumption identifies the same unproved inequality and the same efficiency-based contradiction, so I agree with the reader's assessment. A numerical identity check would settle the matter directly. If the identity holds, the central theoretical claim collapses, and the empirical results alone would not support the paper's framing: the headline gains are a few AUROC points, no error bars are reported, and the β = 0.5 hyperparameter is selected on one benchmark-dataset pair (Table 5, Appendix D). Given the internal inconsistency, the REJECT verdict is appropriate; my stress-test does not change it.","tokens_in":16882,"tokens_out":7416,"duration_ms":94743,"concrete_test":"Take a non-singular 3×3 PSD correlation matrix R (e.g., ρ12 = 0.8, ρ13 = 0.2, ρ23 = 0.4) and compute H = 0.5 ln((2πe)^3 det R) from Eq. (6) and S = Σ_i φ(s_i | s-tilde) from Eqs. (7)-(8). If S = H to numerical precision, then Shapley uncertainty is exactly differential entropy, invalidating the claimed distinction and the joint truth of Propositions 3.5 and 3.6.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing theoretical claim is that Shapley uncertainty is a justified, property-satisfying alternative to differential entropy. But under the standard normalization v(∅) = 0, which Eq. (7) implicitly requires and Eq. (8) uses, Shapley-value efficiency gives Σ_i φ(s_i | s-tilde) = h(s-tilde) for the coalition game v(S) = h([s_j]_{j∈S}). Thus the proposed Shapley uncertainty is not a different metric from the differential entropy in Eq. (6); it is that entropy, computed on a kernel-transformed correlation matrix. This makes Propositions 3.5 and 3.6 contradictory: if h(s-tilde) fails Properties 3.2 and 3.3 as Prop. 3.5 claims, then the Shapley sum cannot satisfy them, whereas if the Shapley sum satisfies them, then h(s-tilde) must also. The Appendix A proof of Prop. 3.6 never addresses this identity. It simply asserts, without derivation, that φ(s_i | s-tilde) > φ(g | s-tilde \\ s_i ∪ g) and that φ(s_j | s-tilde) ≥ φ(s_j | s-tilde \\ s_i ∪ g) for every remaining j. These two monotonicity inequalities are the entire content of the proof, yet no argument connects them to the determinant-based entropy game, and no baseline value for the empty coalition is defined. This is more than a gap: the identity is a theorem of Shapley values, so the claimed conceptual separation between Shapley uncertainty and differential entropy cannot be correct as stated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a new uncertainty metric for natural language generation called \"Shapley uncertainty.\" Given n sampled output sentences, it constructs a correlation matrix from bidirectional entailment probabilities, applies a Gaussian kernel to make the matrix positive semi-definite, and models the sentences as a multivariate Gaussian with that correlation matrix. It then defines each sentence's uncertainty contribution as its Shapley value in a cooperative game whose value function is the differential entropy of the Gaussian restricted to subsets of sentences, and defines the total Shapley uncertainty as the sum of these contributions. The paper states three properties (minimal uncertainty, maximal uncertainty, consistency), claims that differential entropy violates the first two, claims that Shapley uncertainty satisfies all three, and reports AUROC experiments on question answering and machine translation across several models and datasets.","tokens_in":17255,"tokens_out":7102,"duration_ms":83911,"significance":"If the central theoretical and empirical claims held, the paper would offer a plausible, threshold-free alternative to semantic entropy, and the breadth of the experimental evaluation is a genuine strength: the manuscript covers many datasets (CoQA, TriviaQA, WMT-14, SQuAD, BioASQ, NQ-Open, SV AMP) and many model families (OPT, Gemma, LLaMA, Falcon, Mistral, Qwen, DeepSeek), and it provides Algorithm 1 plus an explicit ablation for the kernel hyperparameter. However, the central theoretical claim is internally inconsistent: by Shapley-value efficiency, the total Shapley uncertainty defined in Eq. (8) is identically equal to the differential entropy of Eq. (6), so Proposition 3.5 and Proposition 3.6 cannot both be true. The proof of Proposition 3.6 in Appendix A asserts the very inequalities that need to be proved. Because the paper's headline contribution is the claimed theoretical justification of the new metric, this flaw is decisive.","major_comments":[{"comment":"The central claim is internally inconsistent. For the cooperative game v(S) = h([s_j]_{j∈S}) used in Eq. (7), the standard efficiency property of Shapley values gives Σ_i φ(s_i|˜s) = v({1,...,n}) − v(∅). With the implicit normalization v(∅)=0 in Eq. (7), the total defined in Eq. (8) is exactly h(˜s) from Eq. (6). Therefore Proposition 3.5, which says h(˜s) does not satisfy Properties 3.2 and 3.3, contradicts Proposition 3.6, which says φ(˜s) satisfies them: if h(˜s) fails the properties, then φ(˜s)=h(˜s) must also fail them. The appendix never addresses this identity.","section":"§3.3–3.4, Eqs. (7)–(8) and Propositions 3.5–3.6"},{"comment":"The proof does not derive the key inequalities; it asserts them. For Property 3.2, the proof states without argument that φ(s_i|˜s) > φ(g|˜s\\s_i ∪ g) and that φ(s_j|˜s) ≥ φ(s_j|˜s\\s_i ∪ g) for every remaining j. These inequalities are the entire content of the property and are exactly what must be shown from the determinant-based entropy game. Since the total Shapley uncertainty equals h(˜s), these asserted inequalities would imply monotonicity properties for h(˜s) that Proposition 3.5 denies. No derivation connects them to the Gaussian entropy formula.","section":"Appendix A, proof of Proposition 3.6"},{"comment":"The positive semi-definiteness guarantee is proved only for β ≤ 1/(n+1), where n is the number of sentences. The experiments set β = 0.5, which violates this bound for n ≥ 2. The text says the value was chosen because it maintained positive semi-definiteness in the experiments, but that makes PSD an empirically checked assumption rather than a consequence of Proposition 3.1. This matters because the entire construction of the Gaussian entropy in Eq. (6) requires a valid correlation/covariance matrix.","section":"§3.2 and Appendix A, Proposition 3.1 versus the experimental setting"},{"comment":"The running example is not reproducible from the printed material. The column headers are inconsistent with the listed answers (e.g., \"William Shakespeare\" appears in Table 1 with no corresponding answer row), the displayed entries do not obviously follow from Eq. (5) with the stated Gaussian kernel, and no calculation of φ is shown, so the reported values 0.40 and 0.51 cannot be verified. Moreover, if the displayed correlations are used directly, the matrix contains duplicate rows, which would make the Gaussian differential entropy −∞; since the paper claims Shapley uncertainty avoids exactly such degeneracies, the example needs a clear and consistent computation.","section":"Tables 1 and 2"}],"minor_comments":[{"comment":"The caption spells \"Shapley\" as \"Shapely.\"","section":"Figure 1 caption"},{"comment":"The text contains a duplicated article: \"may not satisfy the the above properties.\"","section":"§3.4, Proposition 3.5"},{"comment":"The proof header says \"Sharpley\" instead of \"Shapley.\"","section":"Appendix A"},{"comment":"The value of h(∅) for the empty coalition is never defined; the Shapley formula requires it. The paper should state the normalization explicitly and then address the consequence that Eq. (8) equals h(˜s).","section":"Eq. (7)"},{"comment":"No confidence intervals or significance tests are reported for the AUROC comparisons; some reported differences are small (e.g., 0.811 vs. 0.808), so it is unclear whether they are meaningful.","section":"§4 and Table 3"},{"comment":"The baselines MaxL, AvgL, MaxE, and AvgE from Manakul et al. (2023) are named in Table 3 but not described in the text; a brief definition would help readers interpret the comparison.","section":"§4"}],"recommendation":"reject","confidential_remarks":"The decisive issue is the Shapley-efficiency identity: with the game defined in Eq. (7), the sum in Eq. (8) must equal h(˜s), making Propositions 3.5 and 3.6 mutually contradictory. This is not a local gap that a revision can patch without changing the definition of the metric or the claims. The empirical study is broad and the kernel PSD preprocessing is sensible, but the paper's central theoretical contribution as stated is invalid, and the empirical comparisons rest on a tuned hyperparameter without statistical inference. I recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the claimed theory does not hold up. By Shapley efficiency, the total Shapley uncertainty in (8) equals the Gaussian differential entropy in (6), the very quantity Proposition 3.5 says fails the properties. The two propositions contradict each other, and the appendix proof is a sequence of assertions.\n\nWhat's genuinely useful: the kernelized correlation matrix in (5) is a reasonable way to turn a possibly indefinite DNN-based semantic similarity into a PSD matrix, and the PSD result is proven. The experiments are broad—many models, datasets, and baselines—and the method usually lands near or above the baselines in AUROC. That is more than many papers in this area do.\n\nWhere it falls apart: the central theoretical claim. Equation (7) is the Shapley value of the coalition game v(X)=h([s_j]_{j in X}). The sum in (8) is the total, and by Shapley efficiency it is exactly h(˜s) under the standard normalization v(empty)=0, which the formula requires. So either Prop 3.5 is false or Prop 3.6 is false. The appendix proof of Prop 3.6 never confronts this. It simply asserts that replacing a sentence with a duplicate of the least-uncertain sentence lowers every other contribution, and an analogous claim for the maximal case. These inequalities are the entire proof, and no argument connects them to the determinant of a correlation matrix. The empty-coalition value is undefined.\n\nOther soft spots: β is selected by grid search on TriviaQA with LLaMA2-7B and then held fixed. That is a fitted constant reused across all reported results, so the \"out-of-sample\" framing is weaker than claimed. The main tables have no error bars, and several AUROC differences are small. The illustrative example in Tables 1 and 2 is not reproducible from the stated formula; no code or data are provided.\n\nWho this is for: someone mining the kernel-correlation idea could find it worth a look, but the paper as written cannot be the basis for a citation. A serious referee would spend most of the report on the identity above, and there is no answer.\n\nRecommendation: desk reject in current form. The empirical part might be publishable as a modest incremental study if the theory is either repaired or withdrawn, but the load-bearing contradiction makes this version unsuitable for peer review.","headline":"The Shapley decomposition sums to the very differential entropy it claims to replace, so the two central propositions cannot both hold; the kernel idea is fine but the theory needs to be rebuilt.","tokens_in":17730,"tokens_out":2676,"would_cite":false,"duration_ms":32778,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Shapley-based uncertainty, built from graded semantic correlations between sampled answers, predicts when an LLM is wrong more accurately than semantic entropy and satisfies three properties the paper argues any valid uncertainty metric…","keywords":["Shapley value","uncertainty quantification","semantic entropy","natural language generation","large language models","question answering","machine translation","AUROC"],"falsifier":"Because Shapley values sum to the value of the full set, the total Shapley uncertainty of any answer set equals the differential entropy of the same Gaussian correlation matrix — exactly the quantity Proposition 3.5 says fails Properties 3.2 and 3.3. A numerical check settles it: take a three-answer correlation matrix, add a perfectly correlated duplicate as $g$, compute every marginal entropy difference in equation (7), verify the efficiency sum, and test whether replacing each sentence with $g$ actually lowers the total. No language model is required for this.","tokens_in":16701,"feed_emoji":"🎯","tokens_out":11809,"duration_ms":122718,"temperature":0.7,"pith_summary":"To decide when a language model's answer can be trusted, this paper proposes replacing the binary threshold of semantic entropy with a continuous score computed from all pairwise semantic correlations among sampled answers. The score, called Shapley uncertainty, is built by turning those correlations into a positive semi-definite matrix, treating the answer set as a multivariate Gaussian with that matrix as its covariance, and distributing the Gaussian entropy over individual answers with the Shapley value. The paper claims this metric satisfies three properties — minimal uncertainty, maximal uncertainty, and consistency — that it argues any valid uncertainty metric should have, and reports that it predicts answer correctness with higher AUROC than semantic entropy and several other baselines across question-answering and machine-translation benchmarks. If correct, the result is a threshold-free, theoretically motivated trust signal for black-box LLMs that needs only sampled outputs and an entailment model.","feed_headline":"Shapley scores spot wrong LLM answers better than entropy","feed_subtitle":"A threshold-free score from sampled answers beats semantic entropy on QA and translation benchmarks.","key_machinery":"The load-bearing object is the correlation matrix built from bidirectional entailment probabilities, together with the kernel variant $K(C(s_i, s_j)) = 1$ for $i = j$ and $K(C(s_i, s_j)) = \\beta\\kappa(1 - C(s_i, s_j))$ otherwise, whose role is to guarantee positive semi-definiteness so the matrix can serve as the covariance of a multivariate Gaussian. The Shapley decomposition then averages, over every subset of the other answers, the marginal contribution of each sentence to the Gaussian differential entropy, with the total uncertainty defined as the sum of the elementary contributions. This machinery converts the thresholding problem — deciding whether two answers share a meaning — into a continuous, fully specified score, and it spreads the entropy across dimensions so that one pair of near-duplicate answers cannot drive the metric to negative infinity.","core_discovery":"The paper's central claim is that uncertainty in natural language generation is measured better through the graded correlations between generated sentences than through a hard partition into same-meaning clusters. It defines a correlation entry $C(s_i, s_j \\mid x) = \\frac{1}{2}P(s_i \\Rightarrow s_j \\mid x) + \\frac{1}{2}P(s_j \\Rightarrow s_i \\mid x)$ for each pair of answers, applies a kernel $K(C(s_i, s_j)) = 1$ on the diagonal and $\\beta\\kappa(1 - C(s_i, s_j))$ off it to force the matrix to be positive semi-definite, and then treats the $n$ answers as a multivariate Gaussian with that correlation matrix. Each sentence receives an elementary uncertainty equal to its Shapley contribution to the Gaussian differential entropy, and the total Shapley uncertainty is the sum of these contributions. The paper states that plain differential entropy fails two of its three proposed properties while Shapley uncertainty satisfies all three, and it reports AUROC results showing the new score predicts whether a generated answer is correct more accurately than semantic entropy, predictive entropy, length-normalized entropy, lexical similarity, and confidence elicitation across more than twenty model-dataset combinations.","pith_inferences":["By Shapley-value efficiency the total Shapley uncertainty of an answer set is mathematically identical to the differential entropy of the same Gaussian, so the paper's two axiomatic claims — that plain entropy fails the properties while Shapley uncertainty satisfies them — cannot both stand; a reader can verify this identity directly and decide which side of the proof to discount.","The empirical AUROC gains are separable from the axiomatic proof: even a failed proof would not erase the reported accuracy gains, which are the part of the paper a practitioner actually needs.","The construction is a generic recipe — sample outputs, build graded pairwise similarity, repair positive semi-definiteness, decompose entropy with Shapley values — so the same score could be tested for hallucination detection in code generation, summarization, or retrieval-augmented generation without any change of method.","The Gaussian mapping is an assumption, not a consequence of how text is generated, so the metric's absolute values are uncalibrated; only its ranking behaviour, as measured by AUROC, is directly supported by the experiments."],"forward_implications":["Threshold tuning for semantic equivalence disappears: the only free choice is the kernel scale $\\beta$, fixed at 0.5 in the paper.","On the reported benchmarks the score flags wrong answers earlier, e.g. AUROC of 0.739 versus 0.674 for semantic entropy and 0.515 for confidence elicitation on CoQA with Gemma-7B.","The three properties give a checklist that any future uncertainty metric can be tested against, independent of this particular construction.","Because the method needs only sampled outputs plus an entailment model, it applies to black-box LLMs with no access to logits or hidden states.","The reported gap over baselines widens for smaller models (e.g. +7.9% over naive entropy on NQ-Open with Falcon-7B), so the metric helps most where the model itself is weakest."],"supporting_citations":[{"why":"Introduces semantic entropy, the threshold-based method this paper extends and the main baseline it compares against.","marker":"Kuhn et al. (2023)"},{"why":"Provides the DeBERTa-based entailment model that produces the probabilities P(si ⇒ sj|x) from which the correlation matrix is built.","marker":"He et al. (2020)"},{"why":"Supplies the length-normalized predictive entropy baseline and the single-model sampling approach for estimating NLG uncertainty.","marker":"Malinin and Gales (2020)"},{"why":"Source of the predictive entropy and p(True) baselines that Shapley uncertainty is measured against.","marker":"Kadavath et al. (2022)"},{"why":"Provides the TriviaQA dataset used for closed-book question-answering evaluation.","marker":"Joshi et al. (2017)"},{"why":"Provides the CoQA conversational question-answering dataset used in the main AUROC comparisons.","marker":"Reddy et al. (2019)"},{"why":"Provides the WMT-14 dataset used for the machine-translation evaluation.","marker":"Bojar et al. (2014)"},{"why":"Supplies the SelfCheckGPT baselines MaxL, AvgL, MaxE, AvgE that Shapley uncertainty outperforms in Table 3.","marker":"Manakul et al. (2023)"},{"why":"Supplies the ask-for-confidence (A4C) baseline compared in Table 3.","marker":"Tian et al. (2023)"}],"fun_headline_variants":["Shapley uncertainty beats entropy in spotting wrong LLM answers","Threshold-free Shapley score predicts LLM errors better than entropy","Graded semantics power Shapley uncertainty, outdoing entropy","Shapley-based LLM uncertainty outperforms semantic entropy scores","New Shapley metric tops entropy for trusting LLM responses"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theoretical result rests on one unproven step in the proof of Proposition 3.6: replacing an answer with a duplicate of the least-uncertain answer is assumed never to raise the uncertainty attributed to any other answer, and if that inequality fails, the three properties do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Shapley uncertainty beats entropy in spotting wrong LLM answers","Threshold-free Shapley score predicts LLM errors better than entropy","Graded semantics power Shapley uncertainty, outdoing entropy","Shapley-based LLM uncertainty outperforms semantic entropy scores","New Shapley metric tops entropy for trusting LLM responses"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000228,"raw_usage":{"total_tokens":1460,"prompt_tokens":913,"completion_tokens":547,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":529,"completion_tokens_details":{"reasoning_tokens":460}},"tokens_in":529,"tokens_out":547,"duration_ms":6270,"temperature":1.0,"reasoning_tokens":460,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T12:48:39.334304+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Because Shapley values sum to the value of the full set, the total Shapley uncertainty of any answer set equals the differential entropy of the same Gaussian correlation matrix — exactly the quantity Proposition 3.5 says fails Properties 3.2 and 3.3. A numerical check settles it: take a three-answer correlation matrix, add a perfectly correlated duplicate as $g$, compute every marginal entropy difference in equation (7), verify the efficiency sum, and test whether replacing each sentence with $g$ actually lowers the total. No language model is required for this.","supporting_citations":[],"review_version":1}