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REVIEW 4 major objections 4 minor 44 references

LLM confidence scores violate basic probability rules even when calibration looks fine.

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2026-08-02 11:40 UTC pith:SAHVXPK2

load-bearing objection The structural coherence metrics rest on event inclusions that are asserted, not derived, and fail for models with inconsistent outputs, so the headline probability-violation claims don't hold as stated; the calibration critique and faithfulness/usefulness axes are still worth a look. the 4 major comments →

arxiv 2607.19367 v1 pith:SAHVXPK2 submitted 2026-06-11 cs.AI

Rethinking Uncertainty Evaluation in Large Language Models

Krish Matta , Atharv Naphade , Andy Zou This is my paper
classification cs.AI
keywords confidence estimationcalibrationprobability axiomsstructural coherenceentailment monotonicityconjunction consistencysampling consistencyLLM uncertainty
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper argues that calibration, the standard yardstick for LLM confidence, is too weak: a constant predictor scores perfectly calibrated while carrying no instance-level information, and measures like RMSCE say nothing about whether a model's reported confidences hang together as one coherent probability assignment. To fill that gap, the authors propose the C1 metrics, which check structural coherence (normalization, conjunction consistency, entailment monotonicity), faithfulness (invariance to rephrasing), and usefulness (calibration plus discrimination). Applied to standard estimators, the metrics show that output-based confidences saturate near 1.0 and merely appear coherent, while a sampling-based estimator that spans the full range reveals genuine violations: models assign lower confidence to easier sub-questions 31% of the time, and multi-hop confidence deviates from the product rule by an average of 0.257. Interventions like RLHF and chain-of-thought improve calibration or usefulness without restoring coherence. The authors conclude that current LLM confidence cannot be read as coherent probabilities, and that C1 provides the tools to measure and close that gap.

Core claim

The paper's central claim is that confidence estimates from current LLMs should be held to the standards of coherent probabilistic beliefs, not just calibration, and that when they are, they fail. Treating a model's credence as a probability measure over world-states, the authors show that calibration admits trivially incoherent estimators (a constant predictor) and depends on the evaluation distribution. They instantiate three structural conditions of probability theory as measurable metrics: normalization (confidences over mutually exclusive answers should sum to one), conjunction consistency (confidence in a multi-hop answer should equal the product of confidences in its sub-answers), and

What carries the argument

The C1 evaluation framework, built on a subjectivist probability model in which a model's confidence in an answer is the credence that the answer is correct in the true world-state. The load-bearing pieces are the three structural metrics, each derived from a probability axiom: normalization (sum of confidences over disjoint answer classes is 1), conjunction consistency (product rule for multi-hop questions), and entailment monotonicity (set inclusion implies no decrease in probability). The comparison estimator that makes the violations visible is SliCK, which clusters sampled rollouts by semantic equivalence and reports the fraction of agreeing rollouts as confidence; unlike verbal or logi

Load-bearing premise

The structural metrics treat correctly answering the full multi-hop question as the same event as correctly answering both sub-questions with the gold answers, an event inclusion the paper asserts holds by construction but which need not hold when the model's chosen sub-answers differ from the gold answers.

What would settle it

Take a set of multi-hop questions, compute entailment-monotonicity violations as the paper does, then recompute them using the model's own most-confident sampled answer to the second-hop sub-question instead of the gold answer. If the violation rate drops well below 31% or the conjunction deviations shrink, the headline structural failures are artifacts of the inclusion assumption. A second check: measure the same structural metrics on a deliberately incoherent estimator (e.g., confidences drawn at random but scaled to calibrate on the benchmark) and see whether C1 correctly flags it while RMS

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Calibration error (RMSCE) should no longer be treated as the primary or sufficient measure of LLM confidence; structural-coherence metrics can flag estimators that are calibrated but cannot be interpreted as probabilities.
  • Output-based confidence estimators that saturate near certainty should be considered unreliable for any downstream decision that presupposes coherent probabilities, even when their calibration error looks low.
  • Sampling-based confidence estimators (like SliCK) are diagnostic tools for model incoherence: their wide score range reveals conjunction and entailment violations that saturated estimators mask.
  • RLHF can improve calibration while degrading discriminative power and structural coherence, so alignment recipes should be audited with C1-style metrics rather than RMSCE alone.
  • Chain-of-thought reduces multi-hop overconfidence and improves conjunction consistency but leaves prompt-sensitivity (faithfulness) essentially unchanged, so reasoning helps only some axes of probabilistic validity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A likely source of the 31% entailment-monotonicity violation is the paper's event-algebra assumption: it counts a confidence gap as a violation only if the model's chosen sub-question answer is fixed to the gold answer, but the model may be answering a different—yet still logically compatible—sub-question; re-running the metric with model-dependent answers would probably lower the violation rate.
  • The framework suggests a direct training objective: penalize conjunction-consistency and entailment-monotonicity deviations during fine-tuning; the paper leaves this unexplored, but it is the natural next step implied by calling structural coherence a necessary condition.
  • The distinction between aleatory and epistemic uncertainty in the paper implies that sampling-temperature interventions could alter structural metrics without changing the underlying beliefs, so improvements from decoding changes may be cosmetic rather than epistemic.
  • A practical test of the framework's value: compare abstention or cascading systems built on C1-coherent scores versus calibrated-but-incoherent scores; if the coherent scores do not drive better downstream decisions, the structural metrics' operational importance would be weaker than claimed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes C1, a three-axis framework for evaluating LLM confidence estimates beyond calibration: structural coherence (normalization, conjunction consistency, entailment monotonicity), faithfulness (prompt and generation semantic invariance), and usefulness (calibration, discrimination). The authors instantiate it with verbalized, logit-based, and SliCK estimators on SimpleQA, MuSiQue, and ParaRel, using Qwen-30B-A3B-Thinking and nine additional models. They report that SliCK is well-calibrated (RMSCE 0.251) but exhibits structural probability violations, including a 0.257 conjunction-consistency deviation and a 31.0% entailment-monotonicity violation rate on MuSiQue. The paper concludes that calibration is insufficient and that current LLM confidence estimates cannot be interpreted as coherent probabilities.

Significance. The conceptual critique of calibration is timely and valuable: the paper correctly notes that calibration admits constant predictors and is distribution-relative, and the proposed distinction among structural, faithfulness, and usefulness properties is a useful organizing framework. If the structural metrics were valid, the empirical results would constitute a significant contribution, showing that well-calibrated estimators can still violate probability axioms. The paper also has strengths: it explicitly names Assumption 3.1 (sampling faithfulness), provides detailed appendices with per-model tables, and correctly identifies that SliCK satisfies normalization and generation semantic invariance by construction, using them as diagnostics rather than wins. However, the central structural metrics are invalid as stated, and the headline empirical claims rest on those metrics.

major comments (4)
  1. [Sec. 4.1.1, Appendix A.3.3] The event inclusion defining Entailment Monotonicity is asserted, not derived. The paper defines E_x = {ω : A[x](ω) = ŷ([x])} and claims that for MuSiQue's two-hop structure, E_x ⊆ E_{x2|x1,y1*} 'holds by construction of the dataset.' This conflates gold-level entailment with the model's chosen outputs. ŷ([x]) and ŷ([x2|x1,y1*]) are produced by two separate prompt calls; nothing in the probability axioms requires that a correct full-answer output coincide with a correct second-hop output from a different prompt. A model can return the correct full answer while failing the standalone second-hop subquestion, so the set inclusion is an empirical assumption about the model, not a consequence of coherence. Consequently, the reported 31.0% violation rate and the claim that models 'assign lower confidence to logically easier questions' do not establish a probability-axiom violation; they may si
  2. [Sec. 4.1.1, Appendix A.3.2] Conjunction Consistency has the same load-bearing flaw. The paper asserts that 'correctly answering the full question x is precisely the event A∩B,' where A is correctness on the first-hop prompt and B is correctness on the conditioned second-hop prompt. This is not a definition; it is an empirical identity that fails for separately elicited outputs. A coherent model can assign high probability to the full question being correct while its independently chosen sub-answers are both wrong. The measured mean deviation of 0.257 on MuSiQue therefore cannot be interpreted as a violation of the product rule P(A∩B)=P(A)P(B|A). The metric would be valid only if the events were defined over a single joint generation (e.g., the full answer and its extracted sub-answers), which is not what the experiments do.
  3. [Sec. 4.3, Abstract] Because the two structural metrics above are invalid, the central empirical claims collapse. The abstract states 'models assign lower confidence to logically easier questions 31% of the time' and 'current LLM confidence estimates cannot be interpreted as coherent probabilities'; both rely on the asserted event inclusions. The data can be reinterpreted as quantifying cross-prompt answer disagreement rather than incoherence. The manuscript even acknowledges this possibility in passing for faithfulness (Sec. 4.1.2), but not for the structural metrics where it is equally applicable. The claim that SliCK 'exposes structural probability violations' is not supported by the experiments as designed.
  4. [Sec. 4.1.1, Appendix A.3.2/A.3.3] The manuscript labels the problematic equalities as definitions ('this is not an approximation but a definition of what multi-hop correctness means'). This is not a defensible operationalization. Even for a perfectly coherent credence function c, the events {ω: A[x](ω)=ŷ([x])} and {ω: A[x1](ω)=ŷ([x1])} ∩ {ω: A[x2|x1,y1*](ω)=ŷ([x2|x1,y1*])} are not equal because the argmax outputs in different prompts are not coordinated. The paper would need to define the events over the model's joint output distribution from a single sampling process, or otherwise justify why the separate-prompt events correspond to a single probability measure. No such justification appears.
minor comments (4)
  1. [Appendix B.2/B.3] The main experiments use 'Qwen-30B-A3B-Thinking' (Sec. 4.2), but this model is not listed in Table 1 or in the per-model results in Appendix C, which include Qwen3-14B and DeepSeek-R1-32B but no 30B model. The reader cannot verify the headline numbers (e.g., RMSCE 0.251, conjunction deviation 0.257, 31.0% entailment violations) against the tables.
  2. [Sec. 2/References] SliCK is cited to Gekhman et al. (2024), which is titled 'Does fine-tuning LLMs on new knowledge encourage hallucinations?' The reference may be correct (SliCK appears to be introduced in that work), but the authors should cite the exact paper or provide a dedicated reference for SliCK, as the title is not self-evidently about confidence estimation.
  3. [Sec. 4.3, Fig. 4/5] The figures are referenced in text but not shown in the manuscript body; the captions provide some context, but the actual data underlying the reported means (e.g., distributions of Δ(x)) would improve interpretability, especially given the large reported deviations.
  4. [Sec. 4.4] The sentence 'Model size does not correlate cleanly with most coherence metrics, but Semantic Invariance shows a clear scaling trend' is supported only by a figure in the appendix. Given the small number of models (n=9) and the 200-question subset, the authors should report confidence intervals or effect sizes; the claim is currently qualitative.

Circularity Check

2 steps flagged

Partial circularity: the conjunction and entailment metrics stipulate the event identities they claim to test, so the headline 'probability violations' partly reduce to those stipulations by construction.

specific steps
  1. self definitional [Appendix A.3.2 (Conjunction Consistency); see also §4.1.1]
    "Correctly answering the full question x is precisely the event A∩B ; this is not an approximation but a definition of what multi-hop correctness means. Since A, B, and A∩B are all events in the same probability space Ω, the product rule applies directly."

    In Section 3.1, E_x is defined as {ω : A[x](ω) = ŷ([x])}, where ŷ([x]) is the model's separately chosen argmax answer. The events A and B are defined analogously from the model's separately chosen first-hop and second-hop answers. No probability axiom makes these events equal: a model can answer the full question correctly while giving wrong answers on the standalone sub-prompts, or vice versa. By stipulating that 'multi-hop correctness' is A∩B, the metric assumes the exact identity needed to interpret any difference between ¯c(x) and c(x1,y*)·¯c(x2|x1,y*) as a probability-axiom violation. The measured deviation (0.257) is therefore partly baked into the definition rather than derived.

  2. self definitional [Appendix A.3.3 (Entailment Monotonicity); see also §4.1.1]
    "Answering the full question x requires answering both hops; answering only the second (given the first) is strictly easier. The entailment E_x ⊆ E_{x2|x1,y∗1} holds by construction of the dataset."

    Gold-level entailment does not imply the model-output event inclusion. E_x and E_{x2|x1,y∗1} use different argmax answers ŷ([x]) and ŷ([x2|x1,y∗1]) elicited from different prompts; even a coherent belief model can choose a correct full answer while failing the separately prompted second-hop question. The asserted inclusion—and with it the conclusion that ¯c(x)>¯c(x′) is a probability-monotonicity violation—is assumed 'by construction' rather than derived from the probability axioms. The 31% violation rate can therefore quantify cross-prompt answer inconsistency rather than incoherent probabilities, so the central claim that SliCK 'exposes structural probability violations' is partially constructed by this stipulation.

full rationale

The paper contains no fitted-parameter-as-prediction move and no load-bearing self-citation; its calibration-insufficiency examples (constant predictor, distribution-relative RMSCE), the saturation findings for verbal/logit estimators, and the usefulness metrics are self-contained and would stand independently. The circularity is concentrated in the operative empirical pillar: the claim that SliCK 'exposes structural probability violations' (conjunction deviation 0.257, entailment violations 31.0%) depends on event identities asserted rather than derived. Appendix A.3.2 defines multi-hop correctness as A∩B and Appendix A.3.3 asserts E_x⊆E_x′ 'by construction of the dataset,' but neither identity follows from the probability axioms or from dataset gold entailments once the events are defined, as the paper itself defines them, using the model's separately chosen argmax answers. Thus the structural tests are specified in a way that bakes in the target conclusion that the confidences cannot be rationalized as a single probability measure. This is partial circularity: the headline 'cannot be interpreted as coherent probabilities' reduces, for these two metrics, to the stipulated event identities. Other axes—normalization, faithfulness, usefulness, interventions—retain independent content, and Assumption 3.1 is explicitly flagged rather than hidden, so the paper is not wholly circular. Score 6 reflects that the structural-violation result is central and partially constructed by definition, while meaningful independent contributions remain.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 1 invented entities

The central empirical claims depend on user-chosen hyperparameters (k, T, B, sample sizes) and on several domain assumptions, the most fragile being the event-algebra identities used to define conjunction consistency and entailment monotonicity. The coherent credence function is a stipulated normative construct, not an independently evidenced entity.

free parameters (4)
  • rollout count k=16 = 16
    Chosen for all main metrics; Section 4.4 shows RMSCE drops from 0.436 to 0.339 and semantic-invariance spread increases as k goes from 4 to 16, so headline numbers are k-dependent.
  • sampling temperature T=0.5 = 0.5
    Chosen for generation; Assumption 3.1 (sampling faithfulness) is temperature-relative, and no sensitivity analysis over T is reported.
  • calibration bin count B=20 = 20
    Standard RMSCE binning; bin count affects absolute RMSCE values and hence cross-estimator comparisons.
  • sample sizes (1,500 main; 200 per task in Section 4.4) = 1,500 / 200
    Chosen sample sizes; the 9-model sweep uses only 200 samples per task and is reported without confidence intervals.
axioms (6)
  • standard math Probability axioms (normalization, product rule, monotonicity) are the normative standard for coherent beliefs.
    Invoked in Section 3.1 via Ramsey's Dutch-book argument and Cox's theorem; accepted background.
  • domain assumption Sampling faithfulness: at evaluation temperature, the generation distribution over semantic classes equals the credence function (Assumption 3.1).
    Stated explicitly; needed to interpret SliCK structural violations as model incoherence rather than decoder noise.
  • ad hoc to paper For conjunction consistency, correctly answering the full question is exactly the conjunction of correctly answering the first-hop and the conditioned second-hop (Full = A∩B).
    Stated in Appendix A.3.2; false for models that give the right final answer while missing the intermediate answer.
  • ad hoc to paper For entailment monotonicity, Ex ⊆ Ex′ holds 'by construction' of MuSiQue's two-hop structure.
    Stated in Section 4.1.1 and Appendix A.3.3; false because the model's chosen answers for the full and sub-questions are generated independently.
  • domain assumption All prompts induce events over one shared world-state space Ω.
    Section 3.1 and Appendix A.1; necessary for comparing confidences across logically related prompts.
  • domain assumption LLM-judge semantic clustering is reliable enough (~94% accuracy) that clustering errors do not change conclusions.
    Appendix A.2 cites Lamb et al. 2026; the paper does not propagate clustering-error uncertainty into the reported metrics.
invented entities (1)
  • Ideal coherent credence function c over world-states Ω no independent evidence
    purpose: Normative reference object against which estimator coherence is judged
    Introduced in Section 3.1 as a normative ideal; not directly observable or falsifiable, and whether any LLM admits such a representation is the very question being tested.

pith-pipeline@v1.3.0-alltime-deepseek · 20692 in / 17742 out tokens · 189172 ms · 2026-08-02T11:40:30.715486+00:00 · methodology

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read the original abstract

Calibration is the primary criterion for evaluating LLM confidence, but it is insufficient: it admits trivially incoherent estimators, depends on the evaluation distribution, and does not test the extent to which the estimation can be interpreted as a consistent, underlying probability function. What we actually need is for LLM confidence estimates to satisfy the conditions required of coherent probabilistic beliefs. We formalize these conditions along three axes (structural coherence, faithfulness, and usefulness) and operationalize them as the C1 metrics. Widely used estimators systematically violate these conditions despite appearing well-calibrated: models assign lower confidence to logically easier questions 31\% of the time, and common interventions reducing RMSCE leave structural violations unchanged, suggesting that calibration is orthogonal to probabilistic validity. RLHF and chain-of-thought improve usefulness metrics without restoring coherence. Our results show current LLM confidence estimates cannot be interpreted as coherent probabilities; our framework provides the tools to measure and close this gap.

Figures

Figures reproduced from arXiv: 2607.19367 by Andy Zou, Atharv Naphade, Krish Matta.

Figure 1
Figure 1. Figure 1: C1 Metrics reduce the flaws in Calibration hard question easier than one it logically implies, violates probability axioms while passing calibration tests. We pro￾pose a richer framework grounded in rational belief theory (Ramsey, 1926; Cox, 1946) and the utility engineering ap￾proach of (Mazeika et al., 2025), defining three categories: structural properties (normalization, conjunction consis￾tency, entai… view at source ↗
Figure 3
Figure 3. Figure 3: Normalization deviation |S(x) − 1|. Output-based esti￾mators violate severely (5.055, 4.132); SliCK satisfies exactly by construction [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Conjunction consistency on MuSiQue. Verbal’s low devi￾ation is a saturation artifact; SliCK and logit-based reveal genuine violations. The main experiments use Qwen-30B-A3B-Thinking (Yang et al., 2025), a 30B-parameter mixture-of-experts reasoning model with 3B active parameters, serving as both generation and evaluation model. Generations in which the model declines to answer or exhausts its token limit a… view at source ↗
Figure 5
Figure 5. Figure 5: Entailment monotonicity violations. SliCK: 31.0% viola￾tion rate; apparent compliance of saturated estimators is a ceiling effect [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Prompt semantic invariance on ParaRel. Output-based estimators vacuously consistent; SliCK genuinely sensitive (µ = 0.163) [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Calibration diagrams and confidence distributions. SliCK alone tracks correctness; output-based estimators report near￾certain confidence regardless of correctness. by construction, yet confidence decreases frequently and substantially. These are model-level failures that saturated estimators cannot surface. Faithfulness failures differ by estimator SliCK trivially satisfies normalization, while verbal and… view at source ↗
Figure 10
Figure 10. Figure 10: As rollout count k increases, calibration error (RM￾SCE) converges downward while semantic invariance violations increase, demonstrating that structural violations are a property of the model distribution rather than sampling noise. pends on the evaluation distribution, and is silent about internal consistency. The C1 metrics operationalize a richer evaluation along three axes: structural coherence, faith… view at source ↗
Figure 11
Figure 11. Figure 11: Scaling laws across metrics. Structural consistency metrics improve as a function of model scale while most other properties do not follow this trend. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗

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