Pith. sign in

REVIEW 4 major objections 6 minor 22 references

Measuring Reasoning Utility in LLMs via Conditional Entropy Reduction

T0 review · 4 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Reasoning chains whose answer-span entropy falls over steps are systematically more likely to be correct—flat or rising entropy marks errors.

desk verdict New empirical observation, but the key claim rests on averaged curves and a target-mismatch the paper never addresses. read the letter →

arxiv 2508.20395 v1 pith:JE6CET5A submitted 2025-08-28 cs.CL cs.AI

Xu Guo This is my paper · ORCID
classification cs.CLcs.AI
keywords conditionalentropyreasoningutilitychain-of-thoughtanswer-spanuncertaintyearlystoppingMATHdatasetteacherforcingtrajectory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that the trajectory of a language model's uncertainty about its final answer—computed as conditional entropy over the answer span after each reasoning step—separates correct reasoning chains from incorrect ones. On the MATH dataset, chains that end in correct answers show entropy that decreases over steps, falls faster, and stays below the entropy of incorrect chains, whose entropy is flat or rising. The authors position this as an oracle study: the entropy signal is measured with a separate inspector model under teacher forcing, requires no reward model or training, and could let a system stop early or prune unproductive reasoning. If the pattern holds, it matters because current approaches spend compute on long chains and samples with diminishing returns; a training-free signal during generation would make reasoning pipelines cheaper and less prone to distraction. The paper also reports that incorrect chains are longer on average and that humans narrow the answer space earlier than LLMs do.

What carries the argument

The carrying object is the answer-span conditional-entropy trajectory: for a response split into reasoning steps Z≤k and answer span Y, the metric is the average per-token entropy Ht = −Σ_v p(v | C, y<t) log p(v | C, y<t) over Y, with the context C = [X; Z≤k] expanded step by step. This quantity estimates H(Y | X, Z≤k), the model's residual uncertainty about the answer after k reasoning steps. The trajectory is produced by repeating this computation for k = 0..K and aligning chains of different lengths onto a common step axis with cubic-spline interpolation (linear if steps ≤ 3), so domain-level averages can be compared. Its work in the argument is to translate 'reasoning utility' into a mea

What would settle it

Match correct and incorrect chains for answer-span length and problem difficulty; if the entropy-decrease separation disappears after matching, the signal is an artifact of length or difficulty. Alternatively, an early-stopping rule that halts when entropy rises should preserve or improve accuracy on held-out MATH problems if the trajectory is truly predictive.

Watch

Extended reading notes

Core claim

The central claim is that the conditional entropy of the answer span, H(Y | X, Z≤k), averaged token-wise over the generated answer under teacher forcing, is a proxy for how much reasoning so far has narrowed the space of plausible answers. Computing this quantity at each reasoning step yields an entropy trajectory. Across seven MATH categories and two generator models (GPT-4o and Qwen2.5-32B, inspected by Qwen3-8B), correct chains consistently show decreasing entropy that falls faster and remains below the entropy of incorrect chains; incorrect chains show flat or rising entropy, especially on lower-accuracy categories. The paper further finds that incorrect chains are significantly longer t

Load-bearing premise

The load-bearing premise is that the entropy the model assigns to the answer span it actually wrote reflects the reasoning's true information gain about the correct answer; on wrong chains this is self-confidence, not gold-answer information.

Editorial extensions

If this is right

  • Entropy trajectories provide a training-free, reward-model-free signal for whether a chain is on track, so a generator could prune or stop early during decoding.
  • A simple selection rule follows from the plots: discard chains whose answer-span entropy does not decrease; if all decrease, keep the ones with the steepest negative slope.
  • Because incorrect chains are longer on average, early pruning of flat or rising-entropy chains should cut compute as well as reduce answer distractions.
  • The human-vs-LLM difference suggests LLM decoding could be improved by strategies that narrow the answer space early rather than incrementally.
  • The metric can be computed from the model's own output logits, so it applies to any autoregressive LLM without extra training.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An intervention the paper does not run: halt generation at the first step where answer-span entropy rises, and measure final accuracy against full-chain accuracy. If the entropy signal is causal rather than merely correlated, accuracy should hold or improve while token cost drops.
  • On wrong chains the measured entropy is self-confidence about a wrong answer; a stronger test would compute entropy against a paraphrase or candidate answer set, to check whether 'decrease' reflects genuine information gain or growing overconfidence.
  • The cubic-spline averaging hides per-problem variation; a per-chain early-slope predictor evaluated on held-out MATH problems would tell whether the first few steps alone carry the signal.
  • The same trajectory analysis could be transferred to non-math reasoning (e.g., commonsense or symbolic tasks) where the answer span is less structured, testing whether entropy reduction is a general signature of useful reasoning.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper presents an oracle study of whether the conditional entropy of the answer span, measured under teacher forcing by a separate inspector model, can serve as a training-free signal for reasoning utility in LLMs. Using MATH problems, GPT-4o and Qwen2.5-32B generate reasoning chains, and Qwen3-8B computes token-level entropy over the answer span after each reasoning step. The authors report that averaged entropy trajectories decrease for correct chains and are flat or rising for incorrect chains, that incorrect chains are longer, and that human reasoning shows earlier entropy reduction than LLM reasoning. They propose a pruning heuristic based on entropy slope. Appendices report null results for cosine-similarity and cross-entropy baselines.

Significance. If the central association were established, this would be a useful training-free signal for test-time compute allocation and chain pruning. The oracle design that separates generator and inspector, the parameter-free nature of the entropy measure, and the honest reporting of failed baselines in the appendices are strengths. However, the evidence is currently descriptive: the central association is not quantified, and the estimator's target is the model-generated answer rather than the gold answer, which undermines the information-theoretic interpretation. The proposed pruning heuristic is not evaluated. With additional per-problem analysis and a gold-answer target test, the contribution could be solid and relevant to efficient reasoning pipelines.

major comments (4)
  1. [Section 3, Eq. (1)] The quantity computed in Eq. (1) is not the conditional entropy of the correct answer. The formal target is I(Y;Z|X)=H(Y|X)-H(Y|X,Z) with Y the correct answer, but the estimator teacher-forces over the model-generated answer span Y from R=Z+Y defined in Section 2. For incorrect chains, Y is the wrong answer, so Eq. (1) measures the inspector's confidence in that wrong answer, not uncertainty about the truth. Figure 1 then averages these target-dependent curves separately for correct and incorrect chains, so the separation could be driven by answer-span identity, length, and problem difficulty rather than by reasoning utility. The manuscript itself notes in Appendix B that cross-entropy over the target tokens is often lower on incorrect chains, which is consistent with this confound. Please recompute H(Y_gold|C_k) for all chains, or otherwise show that the result is not an artifact of the
  2. [Section 5 and Figure 1] The central claim that 'conditional entropy that decreases over steps is strongly associated with correct answers' is never quantified. The evidence consists of averaged trajectories after cubic-spline alignment. No per-problem correlation, ROC/AUC, confidence interval, or statistical test is reported. Since the curves are averaged over many problems, small systematic differences or outliers can produce apparent separation. Please report, for example, the distribution of slopes or entropy reductions for correct versus incorrect chains, the AUC for predicting correctness from trajectory shape, and standard errors over problems. This is necessary to support the word 'strongly' in the abstract and Section 5.
  3. [Section 5, pruning heuristic] The paper proposes a concrete selection heuristic: 'prune chains whose entropy does not decrease; if all decrease, rank by the magnitude of the negative slope and keep the top-k.' This is the main practical payoff promised in the introduction, but it is not evaluated. An oracle experiment on the generated chain sets, for example selecting among N sampled chains by entropy slope and comparing accuracy to random selection or self-consistency, would directly test whether the signal is usable. Without such an evaluation, the heuristic remains speculative.
  4. [Section 4, Data Processing] The alignment procedure may itself contribute to the reported trend. Each chain is resampled to a common target step count using cubic splines. Incorrect chains are substantially longer (Section 5 and Table 1), so aligning to a fixed number of steps rescales the time axis differently for correct and incorrect chains. If entropy tends to decrease with context and then saturate, a longer chain aligned to the same number of steps will appear to fall more slowly than a shorter chain. The paper does not report whether the separation survives alternative alignments, such as absolute token index, no interpolation, or conditioning on the number of steps. Please add a robustness check.
minor comments (6)
  1. [Section 3] The symbol Y is used inconsistently. In Section 2, Y is the model-generated answer span in R=Z+Y; in Section 3 it is called 'the correct answer' and then 'ground-truth Y'. Use distinct symbols, e.g., A for the generated answer span and A* for the gold answer, throughout.
  2. [Figure 1 caption] The legend for the dashed human curves uses identical dash patterns ('--' and '--'), so the human correct and incorrect trajectories cannot be distinguished. Clarify the line styles or colors.
  3. [Appendix C] The generated solution is attributed to 'Qwen2.5-34B', but the rest of the paper uses Qwen2.5-32B. Fix the inconsistency.
  4. [Table 1] The columns Kgt and Kllm are used before they are defined in Section 4. Define them in the table caption or introduce them earlier.
  5. [Abstract] The formula E_{v in V}[-p(v|C) log p(v|C)] is a slight abuse of notation: Eq. (1) averages per-token Shannon entropy over the answer span, not an expectation over the vocabulary. Rephrase for consistency with the main text.
  6. [Reproducibility] No code or data availability statement is provided. Since this is an empirical oracle study, releasing the generation and evaluation scripts would substantially aid reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the entropy trajectories are computed before correctness labels are applied, involve no fitted parameters, and the reported failed baselines show the association was not forced.

full rationale

The paper's central claim is an empirical association between teacher-forced answer-span entropy trajectories and the correctness of LLM reasoning chains. The entropy in Eq. (1) is defined from the inspector model's output probabilities over the generated answer span, with no parameter fitted to the correctness labels, and the grouping of chains into correct/incorrect is based on external human answers from the MATH dataset. Thus the observed decrease for correct chains is not forced by construction: flat or increasing trajectories were possible and are reported for incorrect chains, alongside non-separating cosine-similarity and cross-entropy baselines in Appendices A and B. The self-citations (Xu et al., 2025a,b) appear only as background examples of CoT-style reasoning, not as load-bearing evidence for the entropy claim. The Limitations section candidly notes the oracle requirement and the lack of deployment evaluation. The main weakness — that for incorrect chains Eq. (1) measures confidence in the model's own wrong answer rather than I(Y;Z|X) for the true Y — is a construct-validity threat, not a circularity, because the metric is not defined in terms of the outcome and the reported correlation is not a tautology. No step in the derivation reduces to its inputs by construction or via self-citation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper adds no fitted constants. Its central claim rests on modeling assumptions about what teacher-forced entropy measures, on the validity of spline alignment, and on the standard probability factorization, plus the proxy assumption that self-confidence over the generated answer span tracks correctness.

assumptions (4)
  • standard math Autoregressive factorization P(Y,Z|X)=P(Z|X)P(Y|X,Z)
    Used in Section 3 to frame reasoning utility as conditional mutual information.
  • domain assumption Teacher-forced average token entropy estimates conditional entropy H(Y|C)
    Section 3 Eq. (1); assumes model output probabilities under teacher forcing reflect uncertainty about the generated answer span.
  • domain assumption Cubic-spline resampling to a common number of steps preserves trajectory shape
    Section 4 Data Processing; if resampling distorts trajectories, averaged curves may show artificial separation between correct and incorrect chains.
  • domain assumption The generated answer span Y can be treated as the answer for entropy computation, with correctness judged against human gold answers
    Sections 2 and 3; the paper never conditions on the gold answer when computing H, so the metric measures self-confidence rather than information about the correct answer.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Measuring Reasoning Utility in LLMs via Conditional Entropy Reduction." pith.science (2026). https://pith.science/paper/JE6CET5A

@misc{pith2026250820395,
  author       = {Pith},
  title        = {Pith review of: Measuring Reasoning Utility in LLMs via Conditional Entropy Reduction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JE6CET5A}},
  note         = {Machine review of arXiv:2508.20395}
}
read the original abstract

Recent advancements in large language models (LLMs) often rely on generating intermediate reasoning steps to enhance accuracy. However, little work has examined how reasoning utility contributes to the final answer's correctness. Due to the stochastic nature of autoregressive generation, generating more context does not guarantee increased confidence in the answer. If we could predict, during generation, whether a reasoning step will be useful, we could stop early or prune ineffective steps, avoiding distractions in the final decision. We present an oracle study on MATH dataset, using Qwen2.5-32B and GPT-4o to generate reasoning chains, and then employing a separate model (Qwen3-8B) to quantify the utility of these chains for final accuracy. Specifically, we measure the model's uncertainty on the answer span Y at each reasoning step using conditional entropy (expected negative log-likelihood over the vocabulary) with context expanding step by step. Our results show a clear pattern: conditional entropy that decreases over steps is strongly associated with correct answers, whereas flat or increasing entropy often results in wrong answers. We also corroborate that incorrect reasoning paths tend to be longer than correct ones, suggesting that longer reasoning does not necessarily yield better outcomes. These findings serve as a foundation to inspire future work on designing efficient reasoning pipelines that detect and avoid unproductive reasoning early.

Figures

Figures reproduced from arXiv: 2508.20395 by the authors.

Figure 1
Figure 1. Entropy variation across reasoning steps. Sub-fig. (a-b) compare two LLMs on the same set of math [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Cosine-similarity trajectories between each reasoning step ( [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Cross-Entropy variation across reasoning steps for Qwen2.5-32B. Trajectories generally plateau and show [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Entropy variation across reasoning steps for GPT-4o. The overall trends are similar to Qwen2.5-32B: [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 2 canonical work pages

  1. [1]

    online" 'onlinestring :=

    ENTRY address archivePrefix author booktitle chapter edition editor eid eprint eprinttype howpublished institution journal key month note number organization pages publisher school series title type volume year doi pubmed url lastchecked label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block STRING...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

  3. [3]

    Tom B. Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel M. Ziegler, Jeffrey Wu, Clemens Winter, and 12 others. 2020. https://arxiv.org/abs/2005.14165 Lan...

    arXiv 2020

  4. [4]

    Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, Christopher Hesse, and John Schulman. 2021. https://arxiv.org/abs/2110.14168 Training verifiers to solve math word problems . Preprint, arXiv:2110.14168

    arXiv 2021

  5. [5]

    DeepSeek-AI, Aixin Liu, Bei Feng, Bing Xue, Bingxuan Wang, Bochao Wu, Chengda Lu, Chenggang Zhao, Chengqi Deng, Chenyu Zhang, Chong Ruan, Damai Dai, Daya Guo, Dejian Yang, Deli Chen, Dongjie Ji, Erhang Li, Fangyun Lin, Fucong Dai, and 181 others. 2025. https://arxiv.org/abs/2412.19437 Deepseek-v3 technical report . Preprint, arXiv:2412.19437

    arXiv 2025

  6. [6]

    Chawla, Olaf Wiest, and Xiangliang Zhang

    Taicheng Guo, Xiuying Chen, Yaqi Wang, Ruidi Chang, Shichao Pei, Nitesh V. Chawla, Olaf Wiest, and Xiangliang Zhang. 2024. https://arxiv.org/abs/2402.01680 Large language model based multi-agents: A survey of progress and challenges . Preprint, arXiv:2402.01680

    arXiv 2024

  7. [7]

    Dan Hendrycks, Collin Burns, Saurav Kadavath, Akul Arora, Steven Basart, Eric Tang, Dawn Song, and Jacob Steinhardt. 2021. https://openreview.net/forum?id=7Bywt2mQsCe Measuring mathematical problem solving with the MATH dataset . In Thirty-fifth Conference on Neural Information Processing Systems Datasets and Benchmarks Track (Round 2)

    2021

  8. [8]

    Hunter Lightman, Vineet Kosaraju, Yuri Burda, Harrison Edwards, Bowen Baker, Teddy Lee, Jan Leike, John Schulman, Ilya Sutskever, and Karl Cobbe. 2024. https://openreview.net/forum?id=v8L0pN6EOi Let's verify step by step . In The Twelfth International Conference on Learning Representations

    2024

Show all 22 references
  1. [9]

    OpenAI, :, Aaron Jaech, Adam Kalai, Adam Lerer, Adam Richardson, Ahmed El-Kishky, Aiden Low, Alec Helyar, Aleksander Madry, Alex Beutel, Alex Carney, Alex Iftimie, Alex Karpenko, Alex Tachard Passos, Alexander Neitz, Alexander Prokofiev, Alexander Wei, Allison Tam, and 244 oth...

    2024 arXiv

  2. [10]

    OpenAI, Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Aleman, Diogo Almeida, Janko Altenschmidt, Sam Altman, Shyamal Anadkat, Red Avila, Igor Babuschkin, Suchir Balaji, Valerie Balcom, Paul Baltescu, Haiming Bao, Mohammad Bavarian, Jeff ...

    2024 arXiv

  3. [11]

    Hanshi Sun, Momin Haider, Ruiqi Zhang, Huitao Yang, Jiahao Qiu, Ming Yin, Mengdi Wang, Peter Bartlett, and Andrea Zanette. 2024. https://openreview.net/forum?id=348hfcprUs Fast best-of-n decoding via speculative rejection . In The Thirty-eighth Annual Conference on Neural Info...

    2024

  4. [12]

    Amir Taubenfeld, Tom Sheffer, Eran Ofek, Amir Feder, Ariel Goldstein, Zorik Gekhman, and Gal Yona. 2025. https://arxiv.org/abs/2502.06233 Confidence improves self-consistency in llms . Preprint, arXiv:2502.06233

    2025

  5. [13]

    Xuezhi Wang, Jason Wei, Dale Schuurmans, Quoc Le, Ed Chi, Sharan Narang, Aakanksha Chowdhery, and Denny Zhou. 2023. https://arxiv.org/abs/2203.11171 Self-consistency improves chain of thought reasoning in language models . Preprint, arXiv:2203.11171

    2023 arXiv

  6. [14]

    Yiming Wang, Pei Zhang, Siyuan Huang, Baosong Yang, Zhuosheng Zhang, Fei Huang, and Rui Wang. 2025. https://arxiv.org/abs/2503.01422 Sampling-efficient test-time scaling: Self-estimating the best-of-n sampling in early decoding . Preprint, arXiv:2503.01422

    2025

  7. [15]

    Chi, Quoc V Le, and Denny Zhou

    Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, brian ichter, Fei Xia, Ed H. Chi, Quoc V Le, and Denny Zhou. 2022. https://openreview.net/forum?id=_VjQlMeSB_J Chain of thought prompting elicits reasoning in large language models . In Advances in Neural Information Proc...

    2022

  8. [16]

    Zhihui Xie, Jizhou Guo, Tong Yu, and Shuai Li. 2024. https://openreview.net/forum?id=udZKVMPf3S Calibrating reasoning in language models with internal consistency . In The Thirty-eighth Annual Conference on Neural Information Processing Systems

    2024

  9. [17]

    Yige Xu, Xu Guo, Zhiwei Zeng, and Chunyan Miao. 2025 a . https://doi.org/10.18653/v1/2025.acl-long.1137 S oft C o T : Soft chain-of-thought for efficient reasoning with LLM s . In Proceedings of the 63rd Annual Meeting of the Association for Computational Linguistics (Volume 1...

    2025 doi

  10. [18]

    Yige Xu, Xu Guo, Zhiwei Zeng, and Chunyan Miao. 2025 b . https://arxiv.org/abs/2505.11484 Softcot++: Test-time scaling with soft chain-of-thought reasoning . Preprint, arXiv:2505.11484

    2025 arXiv

  11. [19]

    An Yang, Anfeng Li, Baosong Yang, Beichen Zhang, Binyuan Hui, Bo Zheng, Bowen Yu, Chang Gao, Chengen Huang, Chenxu Lv, Chujie Zheng, Dayiheng Liu, Fan Zhou, Fei Huang, Feng Hu, Hao Ge, Haoran Wei, Huan Lin, Jialong Tang, and 41 others. 2025. https://arxiv.org/abs/2505.09388 Qw...

    2025 arXiv

  12. [20]

    Griffiths, Yuan Cao, and Karthik R Narasimhan

    Shunyu Yao, Dian Yu, Jeffrey Zhao, Izhak Shafran, Thomas L. Griffiths, Yuan Cao, and Karthik R Narasimhan. 2023. https://openreview.net/forum?id=5Xc1ecxO1h Tree of thoughts: Deliberate problem solving with large language models . In Thirty-seventh Conference on Neural Informat...

    2023

  13. [21]

    Zhi Zhou, Tan Yuhao, Zenan Li, Yuan Yao, Lan-Zhe Guo, Xiaoxing Ma, and Yu-Feng Li. 2025. https://arxiv.org/abs/2502.00511 Bridging internal probability and self-consistency for effective and efficient llm reasoning . Preprint, arXiv:2502.00511

    2025 arXiv

  14. [22]

    Jiace Zhu, Yingtao Shen, Jie Zhao, and An Zou. 2025. https://arxiv.org/abs/2409.01281 Path-consistency: Prefix enhancement for efficient inference in llm . Preprint, arXiv:2409.01281

    2025

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.