TokUR: Token-Level Uncertainty Estimation for Large Language Model Reasoning
Authors: Tunyu Zhang
Haizhou Shi
Yibin Wang
Hengyi Wang
Xiaoxiao He
Zhuowei Li
Haoxian Chen
Ligong Han
Kai Xu
Huan Zhang
Dimitris N. Metaxas
Hao Wang
Venue: ICLR 2026 Poster (2026)
Keywords: Uncertainty Estimation
Large Language Models
LLM Reasoning
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: We study the dynamics between token-level uncertainty estimation and mathematical reasoning.

Abstract: While Large Language Models (LLMs) have demonstrated impressive capabilities, their output quality remains inconsistent across various application scenarios, making it difficult to identify trustworthy responses, especially in complex tasks requiring multi-step reasoning.
In this paper, we propose a **Tok**en-level **U**ncertainty estimation framework for **R**easoning (**TokUR**) that enables LLMs to self-assess and self-improve their responses in mathematical reasoning.
Specifically, we introduce low-rank random weight perturbation during LLM decoding to generate predictive distributions for token-level uncertainty estimation, and we aggregate these uncertainty quantities to capture the semantic uncertainty of generated responses.
Experiments on mathematical reasoning datasets of varying difficulty demonstrate that TokUR exhibits a strong correlation with answer correctness and model robustness, and the uncertainty signals produced by TokUR can be leveraged to enhance the model’s reasoning performance at test time.
These results highlight the effectiveness of TokUR as a principled and scalable approach for improving the reliability and interpretability of LLMs in challenging reasoning tasks.

Published as a conference paper at ICLR 2026
TOKUR: TOKEN-LEVEL UNCERTAINTY ESTIMATION FOR LARGE LANGUAGE MODEL REASONING
Tunyu Zhang" 1l Haizhou Shi
Yibin Wang' Hengyi Wang' Xiaoxiao He' Zhuowei Li!
Haoxian Chen Ligong Han Kai Xu* Huan Zhang² Dimitris Metaxas' Hao Wang"1
!Rttgers University UIIC AAmzzon Red Hat Al Innovation
ABSTRACT
While Large Language Models (LLMs) have demonstrated impressive capabili- ties, their output quality remains inconsistent across various application scenarios, making it difficult to identify trustworthy responses, especially in complex tasks requiring multi-step reasoning. In this paper, we propose Token-level Uncertainty estimation framework for Reasoning (TokUR) that enables LLMs to self-assess and self-improve their responses in mathematical reasoning. Specifically, we in- troduce low-rank random weight perturbation during L.M decoding to generate predictive distributions for token-level uncertainty estimation, and we aggregate these uncertainty quantities capture the semantic uncertainty of generated re- sponses, Experiments on mathematical reasoning datasets of varying difficulty demonstrate that TokUR exhibits a strong correlation with answer correctness and model robustness, and the uncertainty signals produced by TokUR can be leveraged enhance the model's reasoning performance test time. These results highlight the effectiveness of TokUR as a principled and scalable approach for improving the reliability and interpretability of LLMSs in challenging reasoning tasks. The source code is avaliable at https://github.com http::/ggthubb.oo//a/--g--e -ang-ML-LLaTTokUKR
INTRODUCTION
Large Language Models (LLMs) have demonstrated remarkable capabilities in various reasoning tasks (Wei et al., 2022a; Wang et al., 2022; Chung et al., 2024; Guo et al., 2025), yet they often struggle to reliably assess the quality of their own responses (Xiong et al., 2023; Tian et al., 2023; Kapoor et al., 2024; Liu et al., 2024; Zhang & Zhang, 2025; Da et al., 2025; Liu et al., 2025). This limitation becomes particularly evident in complex reasoning scenarios where models may generate seemingly convincing but incorrect solutions without indicating uncertainty.
Beyond the dominant body of uncertainty estimation methods that largely focus on short-form question answering (Zhang al., 2023; Yadkori et al., 2024) and classification tasks (Yang et al., 2023; Wang et al., 2024; Shi et al., 2024), two main approaches have been explored for the more challenging setting of sequence uncertainty estimation: (i) Query-level methods (Gal al., 2016; Osband et al., 2023; Hou et al., 2023), despite their solid theoretical foundation, estimate uncertainties u(y with respect to input prompts æ alone, without evaluating the quality of specific generated responses conditioned on those inputs (see Sec. 2)... Besides, these methods require marginalization over the entire output space y; this becomes intractable as sequence length grows. (ii) Response-level methods (Murray & Chiang, 2018; Malinin & Gales, 2021; Kadavath et al., 2022), typically variants of bog-prbbabilities, have shown empirical success but lack strong theoretical grounding (Kuhn et al., 2023). As result, the limitations of the aforementioned methods in capturing response-specific uncertainty hinder the deployment of LLMs in high-stake, reasoning tasks that demand reliable self- asfessssssment.
To address this challenge, we propose a principled framework, dubbed Token-level Uncertainty esti- mation for Reasoning (TokUR), for estimating the uncertainty of generated sequences by aggregating token-level uncertainties based on random low-rank weight perturbation. TokUR introduces
i carefully
'Equal Contribution. 'Correspondence to: Tunyu Zhang <ty.zhang @ @uuu, Haizhou Shi
<haizhou.d shi @ edus, Hao Wang
Work done outside Amazon.

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Published as a conference paper at ICLR 2026
calibrated perturbations to the weights of attention layers, creating an ensemble of model variants that enables principled uncertainty estimation without requiring costly retraining or extensive parameter updates. Building this, we decompose the total uncertainty of each generated token into aleatoric uncertainty (inherent randomness in the data) and epistemic uncertainty (model uncertainty about its parameters), providing theoretically grounded assessment of confidence across the generation process. We then aggregate these token-level uncertainties to evaluate entire reasoning responses, demonstrating both theoretical consistency with established uncertainty principles and practical utility in downstream applications.
Empirically, TokUR enhances LLM reasoning three key aspects: (i) token-level epistemic uncer- tainty effectively identifies incorrect reasoning paths, outperforming baselines across three mathe- matical reasoning benchmarks, (ii) TokUR excels at selecting high-quality solutions from multiple candidates, and (iii) it functions as an implicit reward to guide reasoning. improving accuracy when combined with off-the-shelf test-time-scaling algorithms (Puri et al., 2025). summary, our contributions are:
We introduce TokUR, a training-free token-level uncertainty estimation approach for LLM reasoning through low-rank weight perturbation, providing principled decomposition of uncertainties with proven theoretical properties.
We demonstrate that epistemic uncertainty serve good metric to measure the quality of generated reasoning paths, consistently outperforming conventional confidence metrics across diverse mathematical reasoning tasks.
We demonstrate practical applications of our uncertainty estimation framework: it improves reasoning performance through incorrect path detection, high-quality solution selection, and uncertainty-guider generation.
PRELIMINARIES
In this section, we first introduce the notation used in the remaining sections, and then review the key concepts of uncertainties (See. 2.1) and existing Bayesian LLMs for downstream adaptation (Sec. 2.2).
Notation. In this paper, scalars are denoted by lowercase letters (I)), vectors by lowercase bold-math letters (x), random vectors by lowercase boldface letters (x), and matrices by uppercase boldface letters (X). We use [m] = {1, 2, ... .m to denote the set of consecutive integer numbers from 1 to m. Following convention, we use p for probability, E for expectation, H for entropy, and I for mutual information. Specifically, H[y x] denotes the conditional entropy between random variables y and x. We use H[p(y]x x = x]] to denote the predictive entropy of the output variable conditioned on input æ, with H(p(y(=)) [)]] as shorthand notation when context is clear.
2.1 UNCERTAINTY ESTIMATION OF LONG-FORM GENERATION
Prediction with Bayesian Neural Networks. Bayesian Neural Networks (BNNs) (Neal, 2012;
Hemmändez-Lobutee & Adams, 2015; Gal Ghahramani, 2016; Blundell et al., 2015; Wang & Yeung, 2016; Wang et al., 2016; Lakshminarayana et al., 2017; ;; Wang & Yeung, 2020) predict responses and estimate their uncertainties using the variational distribution g O (D) that approximates the true weight posterior p(0 Given an input sequence x (I1, II) X, the probability of the output sequence y = (y1,.- ... ,, )r) E y is defined as marginalization over the parameters and estimated by Bayesian Model Averaging (BMA) of size M:
p(y)a=)= = / p(y)=; 0) () do Emm p(yla; Obm)), @lm) ((
.M
(I)
Query-Level Uncertainty Estimation. Established techniques of uncertainty estimation (Gal et al.,
2016) mainly quantify the uncertainty of input x (query-level uncertainty) by
Hlp(y [)] Ey-pty mn) x)))
(2)
We provide an implementation of our framework that is compatible with vLLMM (Kwon et al., 2023) for efficient deployment.
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In the context of BNNs (Eqn. 1), the predictive distribution of y is the marginalized predictive distribution over the model parameters, and hence Eqn. 2 is defined as "total uncertainty" (Gal et al., 2016; Depeweg et al., 2017).
A model's uncertainty about a specific input cannot be solely attributed to the randomness of the approximate posterior q D), which is input-agnostic. For instance, when faced with a query "Name a city in the UK?" (Yadkori et al., 2024), even if an infinite amount of data observed (eliminating the randomness of the model parameters), the uncertainty of this question remains high, as there are many correct candidate answers. Hence to distinguish different sources of uncertainty, total uncertainty is decomposed into epistemic uncertainty and aleatoric uncertainty (Gal et al., 2016):
H|p(y/[>]
B) [H[ppy x 0)] +
I(y; 0 x:
Total Uncertainty
Alcatoric Uncertainty
Epistemic Uncertainty
Here, aleatoric uncertainty captures the intrinsic randomness in data and cannot be reduced even with more data observed. In contrast, epistemic uncertainty, defined as the mutual information [y; 0 ()) between y and 0, reflects the model's 's uncertainty about its own parameters, which can in principle be reduced by collecting more evidence. We use (dyyaay (z) defined Definition 2.1 to denote any of the three uncertainties.
Definition 2.1 (Query-Levee Uncertainty). Query-level uncertainty U(y|æ:) is the uncertainty of the predictive distribution p(y [z) given an input query æ. Total Uncertainty (TU), Aleatoric Uncertainty (AU), and Epistemic Uncertainty (EU) in Eqn. 3 are all instances of query-level uncertainty.
Limitations of Query-Level Uncertainty. Using the chain rule for conditional entropy (Cover, 1999), the query-level uncertainty estimation can be decomposed token-by-token as
Σ T
u(y|z)
u(y:ly<..2).
However, the uncertainty term u(y:lyct, 2r) in Eqn. 4 requires marginalization over the random variable yer, which is (i) computationally intractable, and (ii) only reflecting the quality of the input query. Hence, these query-level uncertainties are not proper indicators for evaluating concrete output response y.
2.2 BAYESIAN LARGE LANGUAGE MODELS
Bayesian Low-Rank Adaptation. For a pre-trained network layer with weight matrix Wo, Low- Rank Adaptation (LoRA) (Hu et al., 2022) optimizes the parameters within constrained low-rank subspace. Specifically, the weight update matrix is modeled by AW = BA, where AW E Rm B Rmxr A Re and < min(m, n). The output E m of forwarding the input vector h E RR' nx is then
= Woh + AWh Woh + BAh.
(5)
Leveraging LoRA's parameter efficiency, Bayesian LoRAs (Yang al., 2023; Wang al., 2024; Shi et al., 2024) aim to further integrate BÑN's uncertainty estimation capabilities into LLMS without significant increasing memory complexity. The key idea is to model A and/or B as approximate dis- tributions of the true weight posterior. The asymmetric Bayesianization, exemplified by BLoB (Wang al., 2024) and TFB (Shi et al., 2024), models the elements of A with independent Gaussian distributions while keeping B deterministic. Specifically, we have
9 ( 22) II., q( (Assl Q.y) = II, M.,,)
(6)
where M and Ω share the same shape as and denote the mean and standard deviation of the random variable A, respectively. To estimate this distribution, BLoB jointly trains the mean and covariance through the re-parameterization trick (Wang et al., 2024), while TFB uses a simple training- free maximal variance searching technique by fixing the approximate distribution to the family of low-rank isotropic Gaussian distributions (Shi et al., 2024).
Limited Scope of Existing Bayesian LLMs. Existing Bayesian LLMs have been primarily val- idated in downstream classification tasks of simple single- or multiple-choice problems, where uncertainty estimation is quantitatively assessed via the alignment of prediction confidence and
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accuracy (Yang et al., 2023; Balabanov & Linander, 2024; Wang et al., 2023; 2024; Shi et al., 2024). However, these methods have not yet demonstrated effective generalization to long-form generation TokUR, which estimates token-level uncertainties via
tasks, i.., LLM reasoning. Therefore. weight perturbations, represents an initial step toward extending Bayesian I.IMss to long-form genera- tion, where uncertainty estimation remains largely unexplored and technically challenging.
TOKUR: TOKEN-LEVEL UNCERTAINTY ESTIMATION VIA LOW-RANK WEIGHT PERTURBATION
Sec. 3.1 introduces the key techniques token-level uncertainty estimation. Sec. 3.2 then details how token-level uncertainties can be aggregated for response-leve, uncertainty estimation, and describes the underlying theoretical foundation. Finally, Sec. 3.3 presents our low-rank weight perturbation as posterior approximation. All proofs of propositions can be found in Appendix C.
3.1 TOKEN-LEVEL UNCERTAINTIES IN GENERAL.
Given an approximate posterior (0) D). fixed input query X' and specific output response y = (y1,
yr) .... sampled from the base policy ply x)), we denote the predictive distribution of the next token Y/ produced by marginalization over weights as
P(uelyer, Φ) A (p) Mel., x;; 0)].
(7)
Assumption 3.1 (Stepwise Posterior Sampling). We assume that the weights 0 sampled from the approximate posterior (-((D) are not shared across decoding steps. Formally, the probability of sequence is factorized as
ply II Pyn II. ( ( y.,, 0.11
(8)
instead of adopting the joint formulation
p(y|z) & (p) 0)].
(9)
While both are valid probabilistic models, the joint formulation is incompatible with the autoregressive decoding mechanism LLMs. Hence, we adopt the stepwise formulation in Assumption 3. To validate this assumption, we further conduct ablation study comparing the stepwise formulation in Assumption 3.1 with the joint formulation, and report the results Appendix E.5.4.
Given input and partial output y<u, for the time step t, have the following three uncertainties:
Total Uncertainty (TU) the entropy of random variable y: conditioned on æ and yer
TU(y:leer. x) H VV.,, = = MIEv p(y) Ver. x) log x), (10)
Aleatoric Uncertainty (AU) is the expectation of entropy of random variable y, over the weights 0 sampled from the approximate posterior (-) in Eqn.
AU x)), E [ x; 0)]]
(11)
Epistemic Uncertainty (EU) is the difference between TU and AU:
EU x) TU(y x) AU x) I(y:; 0l),r,
(12)
where V is the vocabulary and all the expectations are estimated with BMA.
3.2 TOKEN-LEVEL UNCERTAINTY FOR RESPONSE-LEVEI UNCERTAINTY ESTIMATION
Definition 3.1 (Response-Leved Uncertainty). Given the token-level uncertainties U(y: yer. x) defined in Eqn. 10-12, we define response-level uncertainty their cumulative sum across all tokens in the output sequence:
x))lx)
Σ
T
(13)
where l denote any of the considered uncertainty measures (TU, AU, EU in Eqn. 10-12).
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Proposition 3.1 (Response-Level Uncertainty Unbiased Estimator of Query-Level Un- certainty). Given an input query a. let y ~ ply|z) be generated sample of length T. Then the response-level uncertainty U (Definition 3.1) is an unbiased estimator of the query-level uncertainty (Definition 2.1), i.e.,
Byyply 2 [[() (y)
(14)
Proposition 3.2 (Token-Level and Response-Level Uncertainty). Given an input query æ. let y --p(y (x) be generated sample of length T. Let (uellyyyy/,a yct, x) denote the token-level uncertainty as defined in Eqn. 10-12, with Uy x)) as the corresponding response-levs uncertainty (Definition 3. I. Our token-level uncertainty is equivalent to the response-level uncertainty when T = 1:
Ügy ax) = (1y l2)).
(15)
The two propositions above provide key connections between Eqn. 13 and existing uncertainty estimation theory (Malinin & Gales, 2021; Ling al., 2024). Proposition 3.1 shows that (y) is an unbiased estimator of the true query-level uncertainty (x)), ensuring its statistical consistency with the ideal formulation. Proposition 3.2 confirms that when the sequence length T 1, e.g.. single-token prediction tasks such as multiple-choice QA (Yang et al., 2023; Wang et al., 2024), the estimator exactly recovers the token-level uncertainty, demonstrating structural consistency. These results support the validity and reliability of our approximation.
A Advantages of Level Uncertainty. Compared Query-Level Uncertainty (Definition 2.1), token- and response-level uncertainties (i) avoid expensive marginalization over sequences (note the difference between y<u in Eqn. 4 and y<t in Eqn. 14) while still (ii) capturing the expected uncertainty conditioned on the generated output response. Moreover, since (y (B<t,, x) depends on the quality of the prefix Yct, (iii) the estimate retains rich semantic information, making it well-suited for entropy-based sequential decision- making (Kuhn et al., 2023; Ye et al., 2025) or hallucination detection (Farquhar al., 2024; Kossen et al., 2024; Ye et al., 2025) in downstream tasks.
3.3 LOW-RANK WEIGHT PERTURBATION AS APPROXIMATION OF WRIGHT POSTERIOR
Suppose that we have an LLM policy p(y]x). To estimate the uncertainty of its output, cast this model into a Bayesian framework by introducing weight perturbations. Due to the established advantages of efficiency, performance preservation of prepeerurbation model, and effectiveness of uncertainty estimation (Shi et al., 2024), we adopt a low-rank structure for the noise added to the model weights. Given a rank-r weight matrix Wo R of neural network layer, we first perform compact Singular Value Decomposition (SVD) (Klema & Laub, 1980):
Wo = U diag(d)VT,
(16)
where d > E RTx1 is the vector of singular values, and U E _2 and V E = both contain orthonormal columns, i.e., U. V V I,-. To ensure computational efficiency, we introduce low-rank noise matrix 6 E Raxr'' whose rank r' & is significantly smaller than the rank of weight matrix, and whose entries are sampled i.i.d. from a Gaussian distribution of standard deviation of Tg. which we refer to as perturbation strength, i.e., Egj (O, of). vi E [n].j [r']. The perturbed weight matrix is then constructed as
W Wo + U'
(17)
where the matrix U contains the top-r' columns of U. This perturbation transforms the deterministic Wo a variational low-rank isotropic Gaussian distribution W (Wang et al., 2024; Shi et al., 2024):
([vee(W)lo,, NVveec W) Eg),
where
M4g vec(Wo)
(18)
Σ, - I
In x
Let 0 denote the collection of all perturbed weight matrices across the model. By assuming the statistical independence among layers, the overall approximate posterior becomes
q (l(o,) = II. g(vee(("))) (W'))og).
(19)
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131
300
0.75
300-
of TOKUR
of 100
AUNOC
Difficatty Levdl
Difficulty Level
Difficulty
Uncertainty of Incorrect Responses
Random (0.5)
AUROC ToKUR (AU, Ours)
AUROC of TOKUR (EU, Ours)
Uncertainty of Responses
(TU, Ours)
Figure 1: Distribution of TokUR's Uncertainty Scores and AUROC across Different Difficulty
Levels, applied to L. ama- - 2-12-11B--- Instrutt. Left: TokUR (AU, Ours); Middle: TokUR (TU, Ours); Right: ToKUR (EU, Ours).
Utilizing the Approximate Weight Posterior ((0)),). Notably, while we leverage the variational
posterior formulation of Eqn. 19 to quantify uncertainty (detailed in Sec. 3.1), we use only the mean weights Wo for decoding of each step rather than BMA as in Eqn. 1. This approach allows for controlled study of the effects of uncertainty estimation itself, separate from the effects of BNNs. For the complete algorithmic description and overview, please refer to Appendix B.
4 EXPERIMENTS
This section presents practical applications of our TokUR for LLM reasoning. For additional experimental results, please refer to Appendix E.
Datasets. We run main experiments on three mathematical reasoning benchmarks of varying difficulty levels: GSM8K (Cobbe et al., 2021) (grade-school arithmetic problems), MATH500 (Light- man et al., 2023) (challenging high school/college mathematics competition problems), and 5,000- example subset of DeepScaleR (Luo et al., 2025) (high-difficull problems from diverse sources). For these complex math problems, LLMs often need perform multi-step reasoning (Wei et al., 2022b; Yao et al., 2023; Zhou et al., 2023) to reach the final answer. These tasks inherently involve long-form generation, therefore well-suited for evaluation of uncertainty estimation methods.
To assess the generalization of TokUR beyond mathematical reasoning, we further evaluate TokUR on five non-math long-form generation tasks, spanning logical reasoning, code generation, and truthfulness evaluation. For logical reasoning, we use three tasks from Reasoming Gym (Stojanovski al., 2025): Zebra Puzzles, Leg Counting, and Color-Cube Rotation. For code generation, we evaluate on the HumanEval (Chen, 2021) benchmark, a widely adopted standard for functional code synthesis. For truthfulness, we use the FactScore (Min al., 2023) dataset, which measures factual consistency by decomposing generated outputs into atomic facts; we follow prior work and use GPT 5 5-mini as both the fact annotator and judge.
Models. We evaluate Llama
TokUR using models from two open-source LLM families: (Grattafiori et al., 2024) and Qwen
and 1.-1-BBInnttrue 1.-5-BBInnstru and 2. 5- Inntrrutu (Team et al., 2024). These models represent recent advances in open-source instruction tuning and provide a practical balance between capability and efficiency. Their differing model scales and architectural families further enable examine the consistency of uncertainty estimation across both model sizes and model types.
Implementation of our TokUR. We estimate token-level uncertainties by applying random pertur- bations as in Eqn. 17 to the query and key weight matrices (WQ, W (Vaswani et al., 2017) in all the attention layers of LLMs (Hu et al., 2022; Yang al., 2023; Wang et al., 2024; Shi et al., 2024). For more details, please refer to Appendix D.I.
4.1 Do TOKUR'S UNCERTAINTIES ACCURATELY REFLECT RESPONSE QUALITY?
This section assesses if our TokUR's uncertainties reflect response quality math reasoning tasks.
4.1.1 TOKUR'S UNCERTAINTIES AND QUESTION DIFFICULITY
Experimental Setting, To better understand the relationships among uncertainty estimates, question difficulty, and their ability to distinguish correct from incorrect responses, sample subset of

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math questions from math-orz (Hu et al., 2025). A question's difficulty level is determined by the number of failed attempts out of 10 when using the Qwen2. 5-3----B--- 1-33---ntt:... ruct model. A difficulty level of O means the model solved the question every time, while a level of 10 indicates it failed on every attempt. We sample 500 questions per difficulty level, yielding a 5,500--uesttion dataset. We then prompt 1.1 ama-3. 2 B- Insttrtrr ruct. solve each question with greedy decoding and apply TokUR to compute uncertainties for both correct and incorrect responses across difficulty levels. Fig. 1 summarizes the results.
Results. TokUR's uncertainty estimates remain positively correlated with question difficulty: for all three types of uncertainty (AU, TU, and EU), incorrect responses consistently exhibit higher uncertainty than correct ones across difficulty levels. In terms of discriminative power, AUROC values are consistently above random (0.5), confirming that TokUR provides useful signals for distinguishing correct from incorrect reasoning. Yet, AUROC tends to decrease as difficulty increases, especially in the mid-to-high range (levels 7-9), showing that uncertainty estimates become less reliable at separating outcomes on challenging tasks. Interestingly, the AUROC score shows slight increase at the highest difficulty level (10). This is likely a result of the imbalanced data distribution (mostly incorrect), where the model consistently produces high uncertainty, which causes misleading high metric value.
4.1.2 TOKUR FOR INCORRECT REASONING PATH DETECTION
Experimental Setting, The preliminary study demonstrates that our TokUR's uncertainty esti- mation can reflect the quality of generated responses, with lower uncertainty generally associated with better outputs. One important application of uncertainty estimation is hallucination detection in LLMs (Farquhar et al., 2024; Kossen et al., 2024; Ye al., 2025). In this context, we treat uncertainty as a scoring function to identify hallucinated (incorrect) responses for long-form reasoning tasks. We adopt three metrics: Area Under the Receiver Operating Characteristic Curve (AUROC), Area Under the Precision-Recht
Curve (AUPRC), and T0p-50% ACC (ACC*) (Farquhar et al., 2024; Ye et al., 2025; Hanley & McNeil, 1982; Boyd et al., 2013). AUROC and AUPRC measure the overall power of uncertainty scores in distinguishing correct from incorrect responses. In addition, report Top-50% ACC, defined as the accuracy of the top 50% samples ranked by the corresponding score. This metric reflects the model's ability prioritize higher-quality generations under fixed budget. We repeat the experiments with three different random seeds to obtain the mean and standard deviation across runs.
Baselines. We systematically categorize our baselines into two distinct types: (i) those relying solely on the LLM's internal signals, including the most recent Self-Certainty (Kang al., 2025), Deep Think With Confidence (DeepConf) (Fu et al., 2025), LLM-Check (Stramanan et al., 2024),
Degree Matrix (Ugee and UDgg) (Lin al., 2023), and INternal States for hallucInation DEtec- tion (INSIDE) (Chen et al., 2024a), as well as classic P(True) (Kadavath et al., 2022), Predictive Entropy (PE) (Malinin & Gales, 2021), Log-Likelihood (LL) (Murray & Chiang, 2018), and (ii) those leveraging external signals, such as auxiliary Natural Language Inference model (He et al., 2020): Semantic Entropy (SE) (Kuhn et al., 2023), and Shifting Attention to Relevance (SAR) (Duan et al., 2024). Note that, apart from the five baselines with underlines (Kang et al., 2025; Fu et al., 2025; Kadavath et al., 2022; Malinin & Gales, 2021; Murray & Chiang, 2018), the others were originally designed for query-level de-hallucination in short-form QA tasks and are therefore not directly comparable to ToKUR; we include them for completeness (see Appendix D.4 for details).
Results. As shown in Table 1, our proposed TckUR consistently outperforms baselines across AU- ROC, AUPRC, and ACC*. For example, on -m
ToKUR (TU) achieves an AUROC of 80.64% % and an AUPRC of 56.79% on MATH500, clearly surpassing all baselines. 8B Inst the improvements are also substantial: TokUR (EU) attains 82.86% % AUROC and 81.35% AUPRC on MATH500, establishing new state-of-the art per- formance. These results highlight an important insight: TokUR provides reliable and scalable uncertainty estimation framework, achieving strong performance without relying on external signals. For results of Qwen models, please see Appendix F.2.
On the larger Ll ama
4.1.3 EXPERIMENTS BEYOND MATHEMATICAL REASONING
Experimental Setting. In this section, we evaluate the generalization ability of TokUR on domains beyond mathematical reasoning. For verifiable tasks such as Logical Reasoning and Code Generation,
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Table 1: Performance of Uncertainty Estimation Methods for Incorrect Reasoning Path Detec- tion. AUROC, AUPRC, and ACC" are all reported as percentage (%). "ISO?" indicates whether the method utilizes Internal Signal Only for uncertainty estimation. We include the accuracy of CoT (i.e., greedy decoding with Shain-of-Though prompting) in the first row for reference. Boldface and underlining denote the best and the second-best performance, respectively.
Method
ISO!
GSM8K
DeepScaleR
AUROC
AUPRC
ACC
AUROC
AUPRC
ACC
AUROC
AUPRC
ACC
1ama--322115
3.2-18-Tn88
COT (Lower-Bound)
25.60=0000
4443zw00
14,25au0
SE
47.2 17.29
2:.7
24.132442
50.641-44
45.09
42.62.
46.30
12.
12:58
SAR
44. 44.57:3301
24,031251
21:07::16
50.28%1047
43.95.077
43.14g1 59.62.1m3
12. 34
11.14:0
Use
48.75.105
25.79
25.201p03
49.05 iM
60.02..44
1836
13.77_0.
14.23
Une
X
60.57:2331
36.32 56.322255
30.93:2004
66.60)20.5
75.72
71 21.99.2.
56. 58
18. 18.04
16.50g203
P(True)
54.38=1.20
26.39
111
56.6419944
48. 22:22
48.92:uu
59,583943
17.48-025
17.52=403
LLM-Cheek
16.41.114
27.01
11.3
59.54, 61.29
45.7610
14.55.022
17.30
INSIDE
55.71
28.8 05
29 20
53.66:2099
46 25
45 195
54,73
15 50
16. 36,30
PE:
57.08.1044
26.881116 14
31 199
71.21,198
61.66 61.61_o0
$9. 19.85.m0
S6.0911
14.74_or3
17.33
25.88+0057
53.8450
13,93=023
16,83=004
Celt-Certaitty
71.17:100
48.37+a00
38. 13:5ml
+3.4
68.38::310
61.38-uw
71.93
33.81=006
21.76
DeepConf
11.
46.001.11
39.87
75.70
69.72
62.77 ium
71.65
29 (9.99..1 pous
22.00
ToRUR (TC. Ours)
80.64:029
56.79:a74
44.67
70.29 75.07-005
83.55:am3
47.56
25.71:003
ToKUR (AU), Ours)
80.61 1072
56.73,975
44.67 18.6
75.03100
02:21
83.52.00
47 p0ms
TokUR (EU, Ours)
79,74
56.646944
44.13-088
71.79
66. 16.40=mmm mc
19.74
82.87
46.76 76:6.3
25.52-0
Llana-3
10..1-000 1111----nneee I---nstructt ruct
CoT (Lower Bouun))
48.60 200m
85, 85.69
24.86,100
SR
62.93
55.21g
55.73 15873
55.6113%
87. 16:6 11
86. am
67. 67
35. 18:1m
35.55:00
SAR
69 69.42.1.
63.741308
19.20111M
89.24 60.161233
87.99
73.0
42.5 22.89_
50.234222
49.48
47.47:2245
84 :4:60:330
84. 87:17
50. 50.16299
25 08:e1s
25.48:1059
Une
58.62
57.69
13.47:114
67.22=1.06
92. 32.24
92.62:10s
59. 142033
32 12.6-1a04
29.75
PrTriee)
33.4
36.051015
41.94;001
82.19
82.77
18:05_00 33.641010
16.23
LLM-Check
37.41
39.6921100 i1
13
73.98:19
37:30
93:23
55.421929
26 46
28.37.20.)
INSIDE
62.941177
55.061311
$7.35 1m
58.8612211
87 44
88, [uw)
67.05...
13.8
34.131100
PE
$7.98:2.a
49,72:054
=3.07=00k
74.03+mm
93. 33:37
93. 23:m
25,90
26 80
28 62
55.36:2144
47.24
=1.07=004
12.2
23.64:310
92.46:100
52.82
24.48=m13
26.85
Self-Certainty
76.41
76.221a07
69.07
801600
95. 65.65 sun
96, 26 pun
76.72..08
56 15.
39303
DeepConf
71.86 86:28:0
69.57
15
83.30:a077
96.23:-m0
96.5::mm
73.05:0
48 48:76:010
37.48-20
ToRUR (TV. Ours)
82.47.0
79.6221911
74.00).mm
95.5 95.53.am i1.01
_5.3
68.25_oml
43.91
TokUR (AU, Ours)
82.434ana
79,56+233
74.00=
30.97:
95. 32:522
95.49.0m
85.3
63 60.20
43,89:40
TOKUR (EU, Ours)
82.86:144
81.35::144
72.40
18.3
94.91
94.67
84.92
65.57=6.3
43.892022
Table 2:
Performance of Uncertainty Estimation Methods on Logical Reasoning Tasks. We report AUROC, AUPRC, and ACC*, all reported as percentage (%), for three logical reasoning tasks from Reasoning Gym. Boldface and underlining denote the best and second-best performance.
Zehru Puzzles
Leg Counting
Cube
Method
AUROC
AUPRC
ACC*
AUROC
AUPRC
ACC
AUROC
AUPRC
ACC+
CoT (Lower Bound)
33.67
35 35.00=icmm
26.00.1:o
PE
44,702mm0
21.762am0
24,002200
28.87
27.44zus
26 26.00
46.73
46.73
28. 28.75
22. 12.000::0000
LL
42.15
21.02
37.92 22.00:ntte
17.10=a0s
32.67=114
46. 46.36-cmm
26.01
24. 24.00:ho00
Self-Certainty
47.77+tmoe
22.95-hare
30.00=m0e
50.46-2030 so
32.88-037
14:00=mmm
50.62:0000
30.80=2000
00:0000000
DeepConf
42.41+tmm
21.07++mm
26.00-mm
42 42.48--mm
28. 18.96-o1s
30.00-amm
49 49.01
24.41-tomm
24.00 24.00-Hoo0
TukUR (TU, Ours]
21.38+t22
41.42+o3s
39.33=11
69.58=220
16:00.
48 18
38-33
11:14--
31.33-t111
TokUR (AU, Ouns)
71.66-t-a3
41.71+aa
39.33-1a1
69. 69.51-118
47.69
48 48-67-115
56-57-18
30.02++16
28.00++3.00
TokUR (EU, Ours]
71.28tbo3
41,20:th 76
37.33=23|
69.66=251
51.27=1u0
50.67=131
60.08=198
33.71-2101
32,00::3.00
we report AUROC, AUPRC, and Top-50% ACC (ACC*), following the protocol described in
Sec. 4.1.2. For the Truthfulness task, we report Top-50% % Precision (Precision*) based on uncertainty- ranked responses. Due to the increased difficulty of these tasks, all evaluations are conducted using Qwen-2. 5-35--3Bnn-- Inn-innsuuu ruct.
Results. The results are summarized in Table 2 (logical reasoning) and Table 3 (code generation and truthfulness). Across all non- non-mathematical tasks, TokUR consistently ranks as the best or second-best method, demonstrating that it generalizes robustly to diverse long-form generation settings while maintaining strong performance beyond the mathematical reasoning domain.
4.2 CAN TOKUR'S UNCERTAINTIES IMPROVE GENERATION QUALITY?
In this section, we explore the direct application of TokUR to reasoning tasks enhance generation quality. Following previous works (Fu et al., 2025), we apply TokUR to measure the confidence of reasoning traces generated from question and aggregate them via voting to obtain final solution.
8

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Published as a conference paper at ICLR 2026
Table 3: Performance of Uncertainty Estimation Methods on Code Generation and Truthfulness
Tasks. HumanEval results report AUROC, AUPRC, and ACC", all reported as percentage (%) FactScore reports Precision (%). Boldface marks best results.
HumanEval
FactScore
Method
AUROC
AUPRC
ACC
Precision
CoT (Lower-Bound)
74.39:tm00
52.43:0000
PE
60.52+000
81.45=a00
78.05=0.0
41.86+00.0
I.I.
56.42:000
79.48:2000
75.61 2:000
45.03=20.
Self-Certainly
61.48.000
80.27aa00
80.49....0
40.10.0.w
DeepConf
63.60=0000
82.75::a.ne
80.49+0000
41.96++0.9
TokUR (TU, Ours)
63.15
81.59 $1.59920
80, 00:70
60.35
ToKUR (AU, Ours)
60,5010.0
81. 81.75.1..
80,90.1.41
61,53
TokUR (EU, Ours)
60.64:0.77
80.75.10.4
80,49
45.26.10.54
Table 4: Performance of Uncertainty Estimation Methods for Test- Time Scaling, Boldface and underlining denote the best and the second-best performance, respectively.
Clema 3. 2-10- 2-10-I:ettue Instruct
Slama- Saanaa--,, 3.1 111-80---- Instruct
Dataset
Score
Method
N=16
N=64
N=256
N=16
N=64
N=256
LL
WBoN
MajeN
47. 10
47.10aass
54.15
54.11=051
58 38.892035
58.92a031
86.74aus2
86.74206
90.48804
90-48:04
91,00
=1.01=021
Self- Certainty
Maj@N
45:02
52.61am
57.182051
80.02,010
87.25
90.05.0m
WBoN
45.02.
52.65
57.22
a0.02,00
87.25
90.05:00
GSM8K
DeepConf
Maj eN
WBoN
46.72:uu9
46.724019
53,47
53.50:00s
58 45.05.44
58.10au4
86.2 86.24
86 245nes
90.32:a04
34
91.02=m00
90.92:021
(Pass@ 1: (4.43//5..6)
ToKUR (TU, Ours)
Maj eN
50.29
17.18
60.68:244
87.6 67.68
90.67
90 90,96
50.29
$7.22
10.71
87.68
90.98.m3
TokUR (AU, Oun)
M
50. 50,20
+7.21
60. 60.70
87.42=0s5
90 00.60:244
90.90 90.99=03:
WBoN
50.20aaus
57.19
60,78:042
87.42aus
90
91.02a00
ToKUR (EU, Ours)
Maj BN
s0538
56.92:000
59.88:a03
88.06
90.69:u477
91.07
WilNN
50.38
56.891091
59.91 juss
88.06 inst
90.67
91.09.00
LL
Maj@N
WHoN
26.42a0s4
33.28
33.30 10
38.56.u1
38.58
50.92.17
50.92.177
59 76
19.36.um
64.02 64.022071
Self- Self-Ceraintty
Muj GN
20.14
29.12
36.68au13
44.00:1.:
55.56
66:66
WBoN
20.14
29. 29.16a059
36.80aas
44.00=142
55.58=116
62 52
MATH500 (Pass GP 1: 25.60 48.60)
DeepCoaf
Maj @N
WBoN
25.68.1
32.442139
33. 15
37.08
38.52
49. 49.88
88.88
19.74.1
28.22a1.00
64 30
ToKUR (TU, Ours)
MajeN
WBoN
27.064
27.06:9054
33.76-001
33.60
39,20aaus
51.26e135
$1 16.261135
59.441135
19.44
63 86:na
63.84a05
ToKUR (AU, Oun))
Maj
27.00 19:0
33.641016
39. 12
51.16111
59.42.11
m4
WBoN
27.06:
33.48
39.10
$1.16214
59.44
63.92mm0
ToKUR (EU, Ours)
Maj
28.28:1131
35.442079
39.44:a5s
52.40.1399
60:90:053
65 32.2
WBoN 28.28
35.44:105
39.38-a01
$2.40=135
61.04:0s5
65.48:435
In addition, we investigate the possibility of utilizing TokUR in an online manner to dynamically guide the generation process itself. Further details of online method are provided in E.4.
Baselines. We adopt Log-LLkelihhon Ligelihooo (LL) as baseline, given its widespread use proxy for generation quality (Manakul et al., 2023; Rafailov et al., 2023; Chen et al., 2024b). In addition, we compare against Self-Certainty (Kang
al., 2025) and DeepConf (Fu et al., 2025), two recent uncertainty-driive approaches for test-time scaling. As our study emphasizes model self-awareness of the boundaries of its knowledge, we do not include baselines that rely on external reward models (Guan
et al., 2025; Puri et al., 2025; Beeching et al.; Uesato et al., 2022; Lightmann et al., 2023).
Response Aggregation with Uncertainties. We first rank all N candidate responses using one of the scoring methods (LL., Self- Self-Certainty, DeepConf, or TokUR) and retain the top-P% candidates. We then employ two common aggregation strategies: Weighted Best-of-N (WBoN) and Majority Voting (Maj@r @N) (Brown et al., 2024). WBoN performs weighted voting by assigning weights to the retained candidates according to their scores, whereas Maj@N simply selects the most frequent response among them, regardless of scoring.
Experimental Setting. We randomly sample 512 responses for each question in MATH500 and GSM8K with decoding temperature of = 0.8. For each N, we first retain the t0p-10% % of samples ranked by their scores. From this subset, the final prediction is determined using either Maj@N or WBoN. Each experiment is repeated 10 times (sample w/o replacement using offline records).

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Published as a conference paper at ICLR 2026
Results. As shown in Table 4, accuracy consistently improves with larger N across both GSM8K and MATH500. Our TokUR-based selection methods achieve clear gains over all baselines, particu- larly in the low-sample regime (N=16) where they deliver up 3-4 points of improvement. Notably, TokUR (EU) attains the best overall performance on both datasets, with strong advantages in the challenging MATH500 benchmark. In addition, results for Maj@h N and WBoN are similar, indicating that both aggregation strategies are similarly effective once the top candidates are identified.
RELATED WORK
Uncertainty Estimation of LLMs. Uncertainty estimation in LLMs is gaining traction for im- proving model calibration in data-scarce adaptation tasks and for reducing hallucinations in text generation (Liu et al., 2025; Vashurin et al., 2025). One prominent approach is Bayesian Adaptation, which combines Bayesian inference with low-rank adaptation (LoRA) (Hu al., 2022) to approxi- mate weight posterior distributions efficiently, avoiding the high computational cost of full Bayesian modeling (Yang al., 2023; Wang et al., 2024; Shi et al., 2024). To estimate uncertainty in generation, two main lines of work have emerged. The first focuses verbalized uncertainty, where models are prompted to express confidence natural language (Lin et al., 2022; Kadavath et al., 2022; Tian et al., 2023; Kapoor et al., 2024). The second line includes logits-based methods, which estimate uncertainty directly from the model's output distributions ( (Van Der Poel et al., 2022; Ren et al., 2023; Duan et al., 2024; Darrin et al., 2023). In parallel, other approaches aim to refine these estimation strategies. For instance, Malinin & Gales (2021) investigates techniques for estimating epistemic uncertainty in structured prediction tasks, while semantic entropy (Kuhn et al., 2023) captures un- certainty by leveraging invariance in meaning across paraphrases. More recently, Zhang & Zhang (2025) introduces method that leverages the reasoning capabilities of LLMs to enhance uncertainty quantification, using chain-of oiiin-of-thooo of-thought prompting to better reflect model confidence in multi-step tasks. These works complement verbalized and logits-based based methods by offering orthogonal perspectives on how uncertainty can be interpreted and measured. Beyond specific estimation methods, recent studies reassess broader principles of L.LM uncertainty: Kirchhof et al. (2025) argues that uncertainty quantification for LLM agents requires rethinking foundational assumptions, while Devic et al. (2025) highlights the limitations of existing approaches and calls for more human-centered uncertainty interpretation.
Uncertainty for Improving LLM Generation. Uncertainty estimation for improving LLM genera- tion, while not entirely novel, has been predominantly limited approaches based on log-probability or its variants, Self-Certainty (Kang et al., 2025) estimates confidence via KL divergence from a uniform distribution, DeepConf (Fu et al., 2025) aggregates top-K pog-brobasillttes as scores. Beam search (Lowerre, 1976; Sutskever et al., 2014; Freitag & Al-Onaizan, 2017; Xie et al., 2023) selects higher-confidencee sequences by retaining candidates with the largest cumulative probbabbbbilty. UAG (Yin et al., 2024) leverages abrupt log-probability changes to select appropriate demonstrations for in-context learning (Brown et al., 2020). UnCert-CoT (Zhu et al., 2025) alternates between greedy and Chain-of- Ohain-of-TThouhhe Thought decoding based on log-probability scores. Complementing these, Zhou et al. (2025) provides theoretical insights into how internal probabilities relate to self-consistency and reasoning errors in L.Mss. Our work differs fundamentally by estimating token-level uncertainties with rigorous theoretical foundations, representing significant step toward extending Bayesian LLMs to long-form generation scenarios.
Limitations. Our work is subject to several limitations. First, although compatible with efficient deployment frameworks such vLLM (Kwon et al., 2023), repeated weight perturbation sampling during inference still poses efficiency challenges for real-time use. Second, our token-level uncertainty aggregation may miss higher-level semantic logical inconsistencies across multiple tokens or reasoning steps, limiting its utility in complex generation tasks. Finally, the problem of high-variance estimation in our TR remains unresolved, constraining reliability in real-world scenarios.
ACKNOWLEDGEMENT
We thank all reviewers, AC, and SAC for their valuable comments. HW is supported by Amazon Faculty Research Award, Microsoft AI & Society Fellowship, NSF CAREER Award IIS-2340125, NIH grant R01CA29783, and NSF grant IIS-2127918.
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APPENDIX
In Appendix A, we describe the role of large language models (LLMs) in our work. In Appendix B, present the full algorithmic description of our method with low-rank weight perturbation. In Ap- pendix C, we provide detailed proofs for all propositions presented in the main paper. In Appendix D, provide implementation details of the experiments, including:
implementation of our TokUR (Appendix D.1),
dataset details (Appendix D.2),
prompt templates used in LLM reasoning (Appendix D.3),
baseline details (Appendix D.4),
and evaluation metrics (Appendix D.5).
Finally, in Appendix E, we present additional empirical results, including:
preliminary study on the uncertainty distributions produced by TokUR (Appendix E.
additional results of Qwen models (Appendix E.2),
detailed numerical results of the test-time scaling (Appendix E.3),
online test-time scaling of TokUR (Appendix E.4),
an ablation study on different components of our token-level uncertainties (Appendix E.5),
and case study of our token-level uncertainties (Appendix E.6).
A LLM USAGE DISCLOSURE
We used large language models (LLMs) solely to assist with polishing the writing this paper, includ- ing improving grammar, clarity, and readability. The LLMs did not contribute to research ideation, experimental design, analysis, or the generation of scientific content. All technical contributions, claims, and conclusions are the authors' own.
B ALGORITHM DETAILS
Algorithm Low-Rank Weight Perturbation Approximation of Weight Posterior.
1:
Input
2: The base model policy pxy[x);
3: The set of weight matrices to be Bayesianized (W)) k= rank of noise matrix r';
4:
5:
The perturbation strength dg.
6:
for 1 N do U, diag(d), + SVD(())))
7:
Eqn. 16
8:
U' +- the first r' columns of matrix U.
9:
Sample noise matrix e R= Gjj N((,
Perturb the weight matrix: W &- W + +Ue
Eqn. 17 b Eqn. 18
11: Get the weight posterior:
end for
13:
Output: The overall approximate posterior: (()lg)) + Ih ((vec(W*))oo)
C PROOF OF PROPOSITIONS
Lemma C.1 (Definition of Conditional Entropy (Cover, 1999)). Give (y,x) ~ ply, x), the conditional entropy H(y(xx) is defined as
H(ylx)
Σ ((x))
DEX
(20)
= Ex-p(x) Hlyl
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Algorithm 2 Particle Filtering for Inference-Time Scaling (Puri et al., 2025)
1: Input
2: The number of particles N;
3: A reward model f;
4:
A LLM Par and a prompt c.
5:
Initialize N particles {-] ~ pM((-c))
6: 1.
7: while not all particles stop do
Update rewards w {f(2) (1) f( ..... f(11a)) }}
(2)
(N)
8:
9: Compute softmax distribution 0 = softmax(w) (w).
Sample indices {
J/=== Po(j = i) O,.
11:
Update the set of particles as {a
Transition {a++1 PMl(le, 2 1
12:
13:
t f 1.
14:
end while
15:
Output: The set of particles in the end.
Lemma C.2 (Chain rule of Conditional Entropy (Cover, 1999)). Let X and be two random variables, then the conditional entropy of the joint distribution H(X, Y) can be decomposed as:
H(X.Y (IX.Y)= X(X) + XY
(21)
Lemma C.I (Cover, 1999) reveals the relationship between conditional entropy H(y(x) and the entropy derived from conditional probability distributions. Lemma C.2 lays the foundation for estimating the uncertainties of sequences. The two lemmas together give us the following proposition. Proposition C.1 (Decomposition of Query-Level Uncertainty, Eqn. 4). Suppose that we have an inpul sequence a and a model policy p(y(x). The sequence-level uncertainty U(y|ær) (r) can be decomposed loken-by-token as:
U(ylæ) T
,,_,,yyyl ((yrly<t,,2)
(22)
where Uyr x) is token-level uncertainty metric as defined Eqn. 10 Eqn. 12.
Proof. For Aleatoric Uncertainty (AU) and Total Uncertainty (TU) defined in Eqn. 10 and Eqn. 11, both are expressed terms of entropy. Therefore, the decomposition of sequence-level uncertainty can be directly derived using the chain rule stated in the Lemma C.2.
For Epistemic Uncertainty (EU), also called mutual information defined in Eqn. 12, we proceed with the following derivation:
H(p(y(æ)) (x)) H. Ep D) p(yy]2z; 0)] (xoo
(23)
(24)
Σ ( Hoeii 0))
+
(25)
T
Σ
+ E Hppyyl))))
(26)
Finally, based on the definition of mutual information, we obtain:
I(y; (z)) H(ply(=)) (ptel;;
I(y:: x)))))))
(27)
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Proposition 3.1 (Response-Level Uncertainty Unbiased Estimator of Query-Level Un- certainty). Given an input query a. let y ~ ply|z) be generated sample of length T. Then the response-level uncertainty U (Definition 3.1) is an unbiased estimator of the query-level uncertainty (Definition 2.1), i.e.,
By m) [M]) (y)a).
(28)
Proof. Based on Lemma C.I, for the token-level uncertainty Uy yet. x)) defined in Eqn. 10-Eqn. 12, we have
E
E pl
(29)
VerEy
=((y)ly<t.))
(30)
Therefore, the uncertainty of the sequence defined in Eqn. 13:
By yyyplyla (x) [((y]==)] = Ex(y(x) t=1 [(y:lueer.))
(31)
(32)
Σ E Pu
(33)
Σ u(y
(34)
u(y
(35)
where the final step follows from the chain rule of entropy (Proposition C.1).
Proposition 3.2 (Token-Level and Response-Level Uncertainty). Given an input query æ, let y p(y|x) (z) be a generated sample of length T. Let u( /yc,, 2) denote the token-level uncertainty as defined in Eqn. 10-12, with U'y x)) as the corresponding response-leve uncertainty (Definition l). Our token-level uncertainty is equivalent to the response-level uncertainty when T 1:
ū(y) x>) = U(y]l=)).
(36)
Proof. When the sequence length T = 1, based on the definition of uncertainty of sequence in Eqn. 13, we have
T
ū(ylx) Σ
U(y x) (uyy)lyy))))) (z).
This proposition implies that the sequence uncertainty collapses to token-level uncertainty when the output sequence length is 1. reflecting the structural consistency of the estimator.
Proposition C.2 (Approximate Distribution of the Weight W Perturbed by Low-Rank Noise, Equ. 18). Given the weight matrix Wn E mR the low-rank noise matrix e E Raxe'' whose rank significantly smaller than the rank r of Wo. and whose entries are sampled i..d. from Gaussian distribution of standard deviation of og: Eij ~ *N(0, o2), Vi E [n], E [r'), we have the perturbed weighted matrix W as defined in Eqn. 17. The variational distribution g(vec( W) lo,))
defined on the weight matrix W is
q(vec( (W) ((vec(W) W)ppg Ma: Eg); where
thg (o),
(37)
I,'
Σn = 0² I 8
0m-r'
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Proof. We begin with compact SVD decomposition of the weight matrix Wo as described in Eqn. 16:
W Udda
(38)
where d > O E R is the vector of singular values, and U E <Rmxr, V E R are orthogonal matrices. We denote the first columns of U as U E Rmxr' to analyze the updated matrix U in Eqn. 17.
Since each entry in e has zero mean, it is evident that the updated matrix also has zero mean. Consequently, have My = vec(WW)
vee(Wo).
Next, we focus on the proof of the variance En. Gien U' (u, u2, u.)) E Rmxr' and E = (21, E2, :. E E Raxr' as defined above, we have the following properties:
U Σ u.u;
Ip-
(39)
0merr
i=1
x'
vec(U'e = vecl Σ uiE = Σ
(E: 8 u:).
(40)
1=1
t=1
We can now derive the covariance matrix as: Eg = Var[vec[
W)] = Varlvee( + UUEEE = Varlvec(U'e T]
(41)
Var
E 2 u.
Varle, u.)
(42)
Ee, [(e @ (E) @ aw)) - I(es @ uu))EE, I(e, @ 1I
(43)
Ee, [G:,]]] 3 (u,u,) - (Ec, [ci] Ber (ci] s (u.u.)}
(44)
Ip I n 3 (u.u.) = o & u = =0g n 3
(45)
On-r
1=1
D IMPLEMENTATION DETAILS
D.1 IMPLEMENTATION OF TCkUR's TOKEN-LEVEL UNCERTAINTIES
Unless otherwise specified, set the rank of low-rank noise to r' 8, the perturbation strength a, 0.1, and the number of samples per uncertainty estimation to M = 2. For the test-time scaling experiments in Sec. 4.2, we apply length normalization to TokUR to mitigate the bias introduced by varying sequence lengths. In contrast, the effect of length normalization may differ in hallucination detection tasks. To investigate this, we conduct additional ablation studies in Appendix E.5.3, examining the impact of length normalization in that setting. To ensure practical applicability in real-world scenarios, we implement our method seamless integration with vLLM (Kwon et al., 2023).
D.2 DATASETS
Table 5 shows the statistics of datasets in our experiments. These datasets collectively span wide range of difficulty levels, from moderate to highly challenging, covering both elementary numerical reasoning and advanced symbolic mathematical tasks. In addition, the problem domains diverse, including: algebra, geometry, and number theory. Such design ensures that our experiments are comprehensive and representative, facilitating a thorough assessment the model's capability across varied reasoning scenarios.
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Table 5: Statistics of the datasets used in our experiments.
Dataset
Samples Used
Split
Task Type
Language
Level
GSM8K
1,300
Training split
Mathematical Reasoning
English
Moderate
MATH500
500
Full
Mathematical Reasoning
English
Difficult
DeepScaleR
5,000
First 5,000 samples
Mathematical Reasoning
English
Highly Challenging
D.3 PROMPT TEMPLATES
In this work, we use the following prompts published by Meta 2.
Prompt Example
Solve the following math problem efficiently and clearly:
-For simple problems (2 steps fewer):
Provide a concise solution with minimal explanation.
-For complex problems (3 steps or more):
Use this step-by-step format:
## Step [Concise description]
[Brief explanation and calculations]
## Step 2:
[Concise description]
[Brief explanation and calculations]
Regardless of the approach, always conclude with: Therefore, the final answer is: answer I hope it is correct.
Where [answer] just the final number expression that solves the problem.
D.4 BASELINES
We compare our uncertainty estimation approach against several baseline methods:
Log-Likelihood (LLL) (Murray & Chiang, 2018): 9): Mean of token-wise log-pro oog-prbbabilities of the output sequence, representing the model's overall confidence in its generation. Predictive Entropy (PE) (Malinin & Gales, 2021): Mean entropy of the predicted distribu-
tion of each token.
P(True) (Kadavath et al., 2022): Directly queries the model about the correctness of its own output and uses the predicted probability of the token "True", normalized by the sum of probabilities of token "True" and "False", as confidence score.
Self-Certainty (Kang et al., 2025): Quantifies confidence using the KL divergence between th predicted token distribution and uniform distribution over the vocabulary at each decoding step.
DeepConf (Fu al., 2025): Computes confidence scores by aggregating the log bog-prbbabilities of the top-K candidate tokens at each decoding step.
The Degree Matrix (Lin et al., 2023): Utilizes the degree matrix the graph Laplacian of the similarities matrix of responses.
LLM-Check (Sriramanan et al., 2024): We faithfully reproduced the official implementation for comparison.
INSIDE (Chen et al., 2024a): INSIDE a method to estimate query-level uncertainty. We tailored INSIDE to setting by asking the LLM to verify the same response multiple times and then calculating the semantic entropy across these verification attempts.
hhtp:://wwuuggggghec.coo/ssssees 3.2-11BInsstttttvvalll
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Semantic Entropy (SE) (Kuhn al., 2023): We adapted SE to setting by prompting the LLM to verify the same response multiple times and computing the semantic entropy of these verification attempts. While this provides signal of response quality, note that SE requires an external NLJ or embedding model, giving it an inherent advantage compared to method.
SAR (Duan et al., 2024): We adapted SAR to our setting by computing sentence-level SAR scores over multiple verification attempts, following similar procedure SE. While this method provides a meaningful proxy for uncertainty, it requires an external semantic similarity model, which raises fairness concerns compared to our approach, which operates solely with the base LLM.
D.5 EVALUATION PARSING AND METRICS
Parsing. To automate the evaluation of outputs generated by large language models, we design specific prompts (see Appendix D.3) that constrain the model to follow fixed structure and require it to place the final answer within a \box (). Considering that in mathematical reasoning tasks, the same answer be expressed in various forms, standardize all answers into a canonical form before comparison (Beeching et al.). During the evaluation, we assess the correctness from two perspectives: numerical equality and symbolic equality, label each generation as "True" or "False".
Metries. To comprehensively assess model performance in binary classification tasks, we adopt the following metrics: Area Under the Receiver Operating Characteristic Curve (AUROC), Area Under the Precision-Recd Recall Curve (AUPRC), and Top 50% Accuracy (Farquhar et al., 2024; Ye et al., 2025: Hanley & McNeil, 1982; Boyd al., 2013).
AUROC measures the trade-off between true positive rate (TPR) and false positive rate (FPR) at various threshold settings. Formally, for set predictions with associated confidence scores, AUROC is computed
AUROC L/ TPR(FPR (x)) dz,
(46)
where TPR and FPR are defined as:
TPR TP+EN
FPR
FP+IN
AUPRC evaluates the off between precision and recall, which is particularly useful in imbalanced datasets. calculated as:
AUPRC L. Precision (x)) dr,
(47)
where precision and recall are defined as:
Precision
TP TP+FP
Recall
TP
TP+FN
Top 50% % Accuracy evaluates the correctness of the top half predictions ranked by confi- dence. Let N be the total number of predictions and be the set of indices corresponding to the top [N/2] predictions with highest confidence. The metric is defined as:
Top 50% Accuracy = [ST
Σ 8(0 = y1).
(48)
iES
where Tg is the predicted label and ys is the ground-truth label.
E ADDITIONAL EXPERIMENTAL RESULTS
E. PRELIMINARY STUDY: DISTRIBUTION OF UNCERTAINTIES
We conduct preliminary study to examine the relationship between responses' token-level uncer- tainties and their correctness. We generate responses on the GSM8K dataset using a greedy decoding strategy with I. ama -.. 2-12--1--2 Inst ruct, and label each response correct incorrect based
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Table 6: Performance of Uncertainty Estimation Methods for Incorrect Reasoning Path De- tection of Qwen famliy models. AUROC, AUPRC, and ACC" are all reported as percentage (%). We include the accuracy of CoT (i.e., greedy decoding with Chain-of- Thought prompting) in the first row for reference. Boldface and underlining denote the best and the second-best performance, respectively.
MATH500
GSM8K
DeapSr
Method
AUROC
AUPRC
ACC
AUROC
AUPRC
ACC
AUROC
AUPRC
ACC
Owen-2.
CaT (Lower Bound)
64.73.cm
87.72_
35.80 Linoo
PE
68.94-ba27
77.30 63
76.80
76.44
95.06-000
94.46-8.0
66.57=2007
46 46.44-200
48 09:t00
68.42:4mxs
76.82-+079
76.
75.54-o11
94.93-cm
94.92-cm
65
47.37-4-om
Self-Certainty
74.95:tm.c
84,76:a
81.33:2e1
77.22:0000
95.66-112
94.77=2000
74.22=m00
62.55-bo022
52.00:2000
DeepConf
74.39.tm33
83.81
80.532mm:
78.0622001
95.63=mm
95.69-200
73.85
60.75.icuz
51.93:1:om
TokUR (TU, Ours]
$1,48+aa
87.42:dmm
86 26
77. 77.66:20.4 40
95.46-0012
96
96.05-8818
76 26:26:
65.47:000r
53.52:+014
To.eU (AU, Our)
82,55
88.30-1ouss
87.33 mazs
78.81
95.76
96.15
77.52
67.30...am
14.61
TokUR (EU, Outs)
78.96
85,37
84.27:206
70.752081
93.60
94.26:2844
73.33=202
61.0822a3
51.35:bo.0
Qwen- 2.5- 5D Instcuct
CoT (Lower-Bound)
75.07=000
92.15-000
44.93a000
PE
80.29
82.00
17.40
97.35_amm
97. 97.23
60.55
50.61
$2.64.1aa4
LL Certuiiyy
62.19:2..9
80. 19.2244
82.13:
75.37
97.02=m3m
97.08=200
60.23-2.10
50.27:2411
52.43:2014
69.54.1um
86.241.m
85.60).114
80.922.01
97.81=0m
97.69-scm
66.31
56.60.1oo
DeepConf
66.8612ms
83.85
84,53:212
81.94:2415
97,88_100
98.31=200
64.91
56.712b0
55.83:200
TokUR (TU, Ours]
s4.6
93.433am
80. 80.15 94.00:aa4
97.36-0011
97.69-2cm
76.05
11.43-imm
63.89.imm
TokUR (AU, Ours)
84.74+aa
93.68--u17
94.00=a.4
10.00==o to
97.44-0
97.79-61
76.39-910
71.73-1331
64.24+6m0
TokuU (EU, Ours)
81.87.5-1.0
91.68
92.00
74.95 m
96.61
97.23
73.55=caa
68.91
61.47:bm
on an exact match with the ground-truth answer. Considering the class imbalance in the model responses, we construct a balanced subset for visualization. Specifically, we retain all incorrect responses and randomly sample an equal number of correct responses. We compute the TokUR (EU) and TokUR (AU) with our proposed token-level uncertainties in Eqn. 13, and plot the results in the Normalized EU-AU space (Fig. 3). We observe that both TokUR (EU) and ToKUR (AU) show a better-than- chance separation between correct and incorrect outputs. Although some overlap exists, their distribution peaks differ significantly, indicating that uncertainty estimates meaningfully correlate with generation quality.
GSMBK
MATH500
(33449g
H-55
N-32
Hode
5-02
N-225
WH532
10urst
(Ours)
Figure 2: Performance on GSM8K (Left) and MATH500 (Right) when scaling up sample size N at test time of Llama-3. 2-1B-Instruct. Our TokUR (AU, EU, and TU) consistently outperforms the LL baseline, particularly when N is small. Please refer to Table 7 for detailed numerical results.
E.2 UNCERTAINTY ESTIMATION FOR QWEN MODEL
As described in Sec. 4.1.2, we further evaluate the generalization of TokUR by conducting ex- periments with the Qwen models. Table 6 reports results for Qwen-2. S-3B-Instruct and Qwen-2. 1-7B-Instruct, using the same experimental setup as in Table 1. Overall, these results provide strong additional evidence that TokUR 's token-level uncertainty estimates maintain a robust correlation with model accuracy across diverse L.MM families and parameter scales.
E.3 TEST-TIME SCALING VIA UNCERTAINTY ESTIMATION
We provide additional visualization of the test-time scaling results in Fig. 2 While the complete numerical results are reported in Table 4, this new figure offers an intuitive view of how accuracy improves with increasing numbers of test-time samples (N 16, 32, 64, 128, 256, 512).
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Labels
Correct
Incorrect
0.6
0.5
0.3
0.2
0.0
0.3
0.6
EU
Figure 3: Distribution of responses from GSM8K (Cobbe et al., 2021) plotted in the Length Normal- ized EU-AU uncertainty space, as quantified by our token-level uncertainty metrics (Eqn. 13).
All experiments use Llama-3. 2-1B-Inst.uet as the base model. For reference, the Pass baseline accuracy (GSM8K: 44.43%; MATH500: 25.60% ) is also shown as red dashed lines, high- lighting the gains achieved through test-time scaling. Moreover, we provide an extended version of the results in Table 7, which builds on Table 4 to include additional test-time sample configurations.
E.4 TOKUR FOR TEST-TIME SCALING (ONLINE)
One popular approach to improving model performance uses a Process Reward Model (PRM) to score each intermediate step during multi-step generation (Guan al., 2025; Puri et al., 2025; Beeching et al.;; Uesato et al., 2022; Lightman et al., 2023), thereby guiding the model's reasoning path. In this section, explore an alternative: guiding the generation process using uncertainty reward, without relying on an explicit reward model.
intrinsic
Experimental Setting. Particle Filtering (PF) (Puri al., 2025) is an inference-time scaling method for LLM reasoning (details in Appendix B). Building upon this algorithm, we use uncertainty as the score for each particle each step to guide the model's generation process. We set the number of particles to N = 16 and the decoding temperature to r = 0.8. We repeat the experiments with three different random seeds to obtain the mean and standard deviation across runs.
Results. Table 8 shows the results. Compared to I.I,, our TokUR, especially TokUR (EU), yields slight performance gain. Given that guiding generation through stepwise scoring is inherently challenging, consider the lack of significant performance gain from uncertainty estimation to be acceptable. Nevertheless, we believe this experiment offers valuable insights that may inform the future design of process reward models.
Table 8: TokUR as Implicit Reward for Test-Time Scaling (Online), on MATH500.
Intrinsic Reward
BoN
WBoN
L.I.
26.27=205
26.27=201
TokUR (TU, Ours)
27.9311025
28.13
TokUR (AU, Ours)
25.200 2..20
25.13.10.1
ToR (EU, Ours)
28.93:200
29.20
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Table 7: Test-Time Scaling for GSM8K and MATH500. Performance comparison of different methods with varying numbers of test-time samples (N 16 to 512) using Llama-3.2- B---B-Instue as the base model. Methods evaluated include log- likeliko (LL) and three variants of TokUR (TU, AU and EU) with both Maj@N and WBoN strategies. Boldface and underlining denote the best and the second-best performance, respectively.
Number of Sumples N
Dataset
Score
Method
N=16
N=32
N=64
N=128
N=256
N=512
Llama-3.2
3.2-1e-Instruct
LL
WBoN
Maj@N
47.10
47.10suss
50.45:0M
50,45e0m
54,11-09
14.15_005
56.775040
46.72,04
58.89e336
58.921337
59.72=tu0
59.81 g.00
Self-Cortainty
Maj@N
45 45.02.102
48.97
_2.61_u7z
S5.22,uer
$7.18
58.03
WBeN
45.02e0m
48.97s381
52,65=000
55.30sa54
57.22+054
58.10=m00
GSM8K (Pus I: 44,43)
DeepCoof
Maj@N
WBoN
46.72+039
46.72
50.12=071
50.12.0m
53.50=005
16.08.04
56. 10
58.10
58.05=344
59.05..0x
58,97=000
ToKUR (TU, Ours)
WBoN
Maj@N
20.29
50.29
53.72
53.72,077
57.22_..45
17.18_
59. 10
59.21
60.68
60.71
61.31
:1,23
ToKUR (AU, Ours)
Maj@N
50:20100
53.77
58.991001
60.70
61.38
WBeN
50.20
53.77 8MM
17.19
19.13
60.78:
61.31
Maj@N
50.38:a93
52,98:247
$6.92=0050
58. 38.77503s
59, 39,88:+350
60.
TokUR (EU, Ours)
WBoN
50.38
52.98
56.89_.05
58.70 in
59. 19.91
60.85g000
LL
Maj@N
26.4210m1
29.28,.0w
33,28__097
17.041071
38.561375
39.00_noo
WBeN
26.42
29.28
33.30_...0
37.02.0u4
38.58
39 39.00..100
Self-Cortinty
Maj@N
20. 14
23.24
29. 12:11
33.80 so
36 688888
38 60:000
WBoN
20.14.1
23.24 14
n9.
33.82,003
36,80
MATH500 (Pass 25.60)
DeepConf
Maj@N
WBoN
25 18
38.88113
28, 361601
28.06+100
12.44
13,30_110
37.34.1 31
36.00s1
37 37.08
38.52
40.40 nua
08.6000000
ToKUR (TU, Oun)
Maj@N
27 06anns
29.18 In
33,76-n54
37 To
39. 10:770
39, 59.00=:00
WBeN
29.18
33,60
37. 37.60
39. 39.20s
39, 39.40mmto
ToKUR (AU, Ours)
Maj@N
27 06:06
29.08
33,64-w0%
37 22.52
39.12
39,20 20,-00
WBoN
27 06209)
29.08
33,48
37 73
39. 10:00
39, 60:000000
TokUR (EU, Ours)
Maj@N
11.32
31.36:
35.44.mm
38.00
39.44:
60
WBeN
28.28
31.36:118
05,44:007
37.862034
39.38
40.00
Llama-22 3.
Instruct
Maj@N
86.74s0s2
89.16
90.48=044
90.99
91.01
91.00=000
LL
WBoN
86.74s063
89.16
90.48=004
90.99sa36
91 11,00=a22
91,00=000
Self Cemainty
Maj@N
80.02a010
84, 84.13
87.25=00
89: 89.22a0.0
90.05
90.77=00
WBeN
80.02sa30
84.13
87,25
89.214930
90,05
90.77=000
GSMBK (Pass 1: 85.699
DeepCoof
WBeN
Maj@N
86 86.242006
86.24111e
88. 88.74
88.74:0M
90.32:
90,34-044
90.90
90 90.88
90,92+32
90, 30294:326
90002:
91.01 =0000
ToKUR (TU, Ours)
Maj@N
89.72
90.67
91.66112
90,96
91.02
WBeN
87 68
89,72
90.65:044
91 06
90. 90.98
91.02-000
Maj@N
87 p0me
89.59..om
90.60_.m1
90. 1a
_0.93_nno
ToKUR (AU, Ours)
WBeN
a7.42
89.59
90.57:044
91. 91.44g03
90.98
91.02:1w0
ToKUR (EU, Ows)
WBoN
Maj@N
18.0614s
88.06 88,06
89.88
90.69
90.67 n:48
91. 11.19
91.19.03
91.07
91.09
91.02
91.05_ma
Maj@N
10.9211m
55 15.24
$9.36_nxx
62. 1m
64.10
65.00_nox
LT.
WBeN
50.92s1m
55.24
59.46-w7s
80:
64.02
65.00=wwx
Self-Cortaiuty
Maj@N
44.00s11:
48 :8
55.56
60.04
62 62.66
65.40=10x
WBoN
44.0011s:
48.48
55.58_110
59.901051
62.52.a41
64.80_no0
MATT1500 (Pass 48.60)
DeepConf
Maj@N
WBoN
49 29
49 18
54,42s))44 46
55.04
58,22=1000
59, -1.744mmm
60:90s1100
62 10me
63, 13,434
14.30
65.20 65.20_noo
64,80=000
TokUR (TU, Ouns)
Maj@N
31.262136
55.544200
19,44-1131
62. 28
63 86
65.20 60:20=000
WBoN
$1.261116
55.54.om
59,44_1.8
62.32.110.
63.84
65.20_nox
TokUR (AU, Ours)
Maj@N
WBeN
$1.1611
11.161145
55.52.0m
55.52200
59,42__13
59.4 10.44:1..9
62. 02.34g1 16
11
64 La04
63 92
65.60 63.60-000
65.60
ToKUR (EU, Ouns)
Maj@N WBeN
52.40.1.9
57.02:3a
60.90-033
64,24:a83
65 65.32
67.00
52,40=1.9
57.02:am1
61.04:05
64.20
65,48:475
67.00-0.00
E.5 ABLATION STUDY
This section presents an ablation study our token-level uncertainty estimation method using low-rank perturbations. Appendix E. examines the effect of varying perturbation strength og. while Appendix E.5.2 analyzes the impact of different decoding temperatures. In Appendix E.5.3, we investigate the effect of length normalization in the context of hallucination detection tasks. Finally, in Appendix F.5.4, assess the validity of Assumption 3.1 as it pertains to TokUR.
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Table 9: Uncertainties for Incorrect Reasoning Path Detection. AUROC, AUPRC, and ACC* are all reported as percentage (%), where ACC* (%) denotes the accuracy of the Top 50% generations identified by different uncertainty measures. Rows with shading indicate methods without Length Normalization (LN) for uncertainty estimation.
MATHS00
GSM8K
DeepScaleR
Method
AUROC
AUPRC
ACC*
AUROC
AUPRC
ACC
AUROC
AUPRC
ACC
LLama-3.2-11 1.--------I-nttuut
SE
47.29
25.71
24. 24.131442
50.6414.4
45.091072
42.621016
46.30.021
12.94102s
12.5
SAR
44.57
24.03=255
21.07
50.281007
43.24:1099
43.95 43.14:142
12.34
11.14
URree
48.75=1.05
25.79=1.33
25.20=0.33
49.05=0.6
60.02+044 59.62=02:
48.68#024
13.77#029
14.23
UDg
60.57=231
36.32=2259
30.93=004
66. 00.000
75.72+0.3 71.99=0.0
50.88+054
18.04=a63
16,50+a39
P(True)
54.38=1.20
26.39=1.26
27.60=1.18
56.642004
48, 48.222408 48, 48.9220m 59.58=044
17.48=a02
17.52+
LLM-Check
56.412096
27.01=1:2
31.
71.01 61.29 59.542000
55.762048
14.55
17.30
INSIDE
55.71
28.8 82.442 29.20.1433
53.661052
46.03 45,79 45,791125 54.73.032
15.5010.48
16.30
PE
57.08=0.50
26.88=1.05
31.33+0.52
71.21
+ 59.85=0m 46.09
14,74=023
17.33+
LL
=5.41=0.4
25.88 00.87
29.87
58.51 69.01
57. 37.3820m
23.84
13.93=22xx
46.83+
-LN
29.38
54.64
43. 23.73 73.67..0:0
68 881000 60.92.000
82.62.101
45.761002
25,431a0
Self-Certainty
71.17.0.30
48. 1..5
38. 13.11
73.41
68. 38.0000000
61.38.000
71.93.004
33,81 1.08
21.76.104
-LN
23.76::0.34
17.04=0.10
11.20::0.00 34.42+0.00
34.20:0000 31.54:1000
21.33
8.57
4.04
DecpConf
71.77
46.00=0042 39.87 39.87=0044
75.70=m00
69.72=0.0
62.77=000
71.65=0044
29.99
22.00+a04
-LN
25.79=0.54
=7.41
11.47=0.33
28.84
36.22+000
35,23
23.87=001
8.80=6.01
4..0
TokUR (TU, Ours)
57.14go.81
26.92.n00
=1
70.922004
61.322a.3
58 15
56, 26.20
14.79
17.522a5
LN
80.64=0.29
06.79
44.67
75.07
70.29
02.31
83,55+a002
47.56=a04
+
TokUR (AU, Ours)
56.95=0.82 26.81 =0:09
31. 31.60=0.58
70, 70.90+0.05 61.26+0.13
58.87+0.32 56.02=044
14.73+019
+7.47+007
I.N
80.61
56.73=0.75
44.67
20.66
75.03+000
70.22;005
62.21 10:18 83.52:002
47.48:205
25.71 zam
TokUR (EU, Ours) 61 14:64 31.07:11M
33.20::142 65. 15.998 60.02.1022 56.051075
62.10
17.73
19.10
LL
79.7410.21 56.64_041
44. 13..3 71.7910.00
66.401102
59.741100
82.871032
46.761038
15.5
Negative Length
36.2720:4 49.555110
41.872046
65.6920.0
56.722000
56.8720.8
20
35.97
20.4
Llama-3. 1..ma-3.11-- 1-88 Instrrctt
SE
62.93
15.2 55.73
55.61
17.16
86.772101
18 67.
35
SAR
69.42.12.9
63,74 59.20
60.161222
89.241004
87.991081
73.01 1028
42.8910s
37.51
URer
50.23
49 49.4
49.60a224
47.47
84.691089
84.872117
50.16.006
25.08
25,4 1055
U Deg
58.62=0.36
57.69=0.50
53.47=164
67.22+1.06
92.24+055
92.62+0a8
59.14
32.64=043 29.75
P(Truc)
33.41=0225
36.05=0.5 35.33=0.19
41.94=001
82.19
82.77=000
33.64
18.06=au6 +6.23+amm
LLM-Check
57.4110.4
19.69
12.80
73.9810.1
93.371001
93.231000
55.42.027
26.4610.9
28.3
INSIDE
62.94
55.06=3.9
10.3
58.86#2.11
87.44.094
88.21 10:0
67.05
33,83+n42
34
PE
57.98=0.49
49.72=0.54
=3.07
74.03=n01
93.37
93.23=0000
55,90=002
26.80=n16
28.65+aa2
LL
55.3620.49
47. 47.24=00:0 51.07
72.21
92.64
92.462000
52.82+032
24.48=a013 26.85
81.36220.0
28.80=003
92.27
80.032001
95.30
95:082000
84.58:0077
63.92:00x
43.69:a14
Sell-Certainty
26.4
76.222.0.7 69.07
80.60 10.6011.11 95, 05.65100s
96.261009
76.72
$6.15
1920
-LN
21.9410.58
33,57
28.40 18.401100
26.441001
75.431000
17.85
18,801021
15,15
8.1710.8
DeepConf
71.86=0070
69.57=0..4 66.27
83. a3.30
96.23+0002 96.56
73.05
48.76an10 37.48
LLN
23.73
34.08.0044
30.800 10.80..106
30.8610001
77.301000
79.38.100
21.20
15.501011 9.27.1..6
TokUR (TU, Ours)
56.4910.46
48 10.85 52.131.0.5
73.9810.0
93,27100a
93.1310099
14.8610.17
25.9710.1
17.9
82.47
79.62=0033 74.00
81.01
95.531000
95.54.1000
85.33
65.25
2002
43.91 1009
TokUR (AU, Ours) 56.31 #0.47 48.1 =0.54 51.87
73.97
93.26*005
93. 93.13
54.77=0.18
25.90+n13 27.93+am1
- -LN
82,43=044
79.56=0035
74.00=0.99
80.97=0.2
95.52+008
95.49+009
85031+007
65.20=am2 43.89+a0s
TokUR (RU, Ours)
60.9220.46
52.642071
56. 36.1311. 67. 67.92+0.2
92.41 2022
92, 15:441 57.422002
28.3222.16 29165
82.8620042 81.35 72. 12.40 78.311158
94.91 94.67:1077 84,92:0028
65.571045
4328
Negative Length
78. 18.11ah 100.4 23.8 68.80 20
73.6442001
93. 93.362000
93.542000 81.20
S2.12 41157+a16
E.5.1
THE EFFECT OF PERTURBATION STRENGTH og ON UNCERTAINTY ESTIMATION
To investigate the impact of perturbation strength on uncertainty estimation, we conducted a series of experiments under varying og settings, as shown in Fig. 4. First, we computed the average uncertainty estimates (TU, AU, and EU) on samples generated from the GSM8K test dataset using .lama-3.22 3- Instruct. As illustrated in Fig. 4 Left, the model's uncertainty increases steadily with higher perturbation strength. However, once o, exceeds a critical threshold (c.g.. 0.2), a sharp rise in uncertainty is observed. This rise illustrates that the current approximate distribution of the weights has deviated too far from the pre-trained point estimation of the parameters, leading to unreliable uncertainty estimates.
We further evaluate the effect of perturbation strength on downstream task performance. Specifically, we assess how effectively the uncertainty estimates, obtained under different og values,
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0.75
290 TU
0.24
EU
AUROC Score
192
0.6
AU
0.55
03
04
Pertrrattion Strength
Perturbation Strength
Figure 4: Left: Uncertainty estimation with different perturbation strength og. Right: Influence of
perturbation strength on uncertainty-basse AUROC scores.
AU
EU
AUROC Score
0.75
0.60
TU
Temperature
Dacading Tamperature
Figure 5: Left: Uncertainty estimations in different token decoding temperature Right: Influence of token decoding temperature on uncertainty- -ased AUROC scores.
can be used as scoring signals to distinguish between correct and incorrect samples, as described in Sec. 1.2. As shown in Fig. 4 Right, for TokUR (EU), too small 0g does not lead to meaningful improvements log-likelihood, whereas an excessively large o, harms performance all three TokUR variants (AU, TU and EU) by distorting the original semantic content. Based on these findings, we set 0g 0.1 for the experiments reported Sec. 4.
E.5.2 THE EFFECT OF TOKEN DECODING TEMMERATURE: UNCERTAINTY ESTIMATION
During text generation with large language models, the decoding temperature introduces uncertainty into the model's output. In general, higher temperatures lead to more diverse responses. In this section, we investigate the relationship between decoding temperature and uncertainties estimated by our token-level approach. Specifically, we use L Llama-3 2.2 1B Inst ruct to answer questions from the MATH500 dataset under different decoding temperature settings and estimate the average uncertainty of the model's responses.
As shown in Fig. 5 Left, increasing the decoding temperature results in notable rise in token- level Alcatoric Uncertainty (AU) of the model, whereas the Epistemic Uncertainty (EU) remains relatively unaffected. Additionally, approaches across different temperature settings in Fig. 5 Right. These results indicate that varying the temperature does not harm the performance of TokUR, highlighting its robustness to changes in decoding temperature.
report the AUROC scores of various uncertainty estimation
E.5.3 ABLATION STUDY OF LENGTH NORMALIZATION
Length normalization is standard technique for aggregating token-level uncertainty into sequence- level uncertainty (Fu et al., 2025; Kang et al., 2025), as it mitigates the bias introduced by sequence length when evaluating generation confidence. However, described in Eqn. 13, do not apply
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Scaling
(TV)
Mean Accurscy
Mean Recurnacy
Zopeetin Hodd rg (44
Sizzerie io Modding M4gg10
Sisprii Medsling IPajp Soaprius Kodsiing 19224
Zupratie (oitoi
Siazeas Hodding
Modaiing Milel
Jant Hods (MBu))
tia
zie (EU) Improvament over Baseline
ToKUR (AU) Iipprvvmenntt over Baseline
TokUR (TU) Improvement over
Accuracy improvement (1)
Accurnry Improvement (N)
Accurscy
Supeeis
Suppais Vodeling
Suppetun Hoadl rg Mibohi
Siepe sn Modding (Abol)
Sispiis Medsiiig (9227)
Modding M4
Kedciing MUNK
Junt tng (Meg
120
52
sie
512
Uok (AU) - Scaling Efficeency
TokUR (EU) - Sceling Efficiency
TokUR (TU) Scaling Efficiency
EHt serry (impooemme (Sample)
Eniccen engyy perprrmmeett...m Samplel
02-
Mold Impulio More (M yonr restring (4aano
Em
Stow on Hextrine (wib Jear Meaniin PAASE
Cripaine Mesn (B6ae)) jair Made ino (Maueto
-35
Hursera' 156"
Hurber daSare
Figure 6: Ablation of stepwise posterior sampling. Comparison of stepwise vs. joint modeling 1.1 allama-3. 3..m--32211 B2-1B-Insttuu across accuracy, improvement, and efficiency. Stepwise modeling consistently achieves better scaling performance, validating Assumption 3.1.
normalization when computing TokUR. To assess the impact of sequence length on uncertainty estimation, therefore conduct an ablation study on length normalization.
Experimental Setup. We investigate the effect of length normalization on incorrect reasoning path detection across three datasets (MATH500, GSMSK, and DEEPSCALER), following the same settings as in Table 1. We compare TokUR with and without Length Normalization (LN), along with representative basclines. addition, we introduce a naive baseline, Negative Length, which uses sequence length alone as confidence signal.
Results. As shown in Table 9, the impact of length normalization varies significantly across methods. For both LLL and TokUR, normalization consistently reduces AUROC and AUPRC, indicating that raw sequence length introduces favorable bias that benefits uncertainty aggregation in de-hallucination tasks. This observation further reinforced by the strong performance of the Negative Length baseline, which alone achieves competitive results across all datasets. In contrast, Self- Self-Certainty and DeepConf show clear gains with normalization (e.g., Self-Certainty improves from 23.76 to 71.17 AUROC MATH500), suggesting that normalization is essential for stabilizing their performance. Overall, these findings reveal that the role of length normalization is highly method-dependent.
E.5.4 ABLATION STUDY OF STEPWISE POSTERIOR SAMPLING
To examine the validity Assumption 3.1, we perform an ablation study comparing step- wise posterior sampling against the joint posterior formulation. Concretely, evaluate
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Published as a conference paper at ICLR 2026
test-time scaling MATH500 dataset, using TokUR with both stepwise and joint modeling Llama-3. 3...-B.--.------- Ins-IInnttrttt while keeping all other settings consistent with Table 4.
As shown in Fig. 6, stepwise modeling consistently outperforms joint modeling across all uncertainty measures (AU, EU, and TU). Specifically, stepwise sampling achieves higher mean accuracy and the baseline, while also demonstrating superior scaling efficiency with increasing numbers of samples. These results provide strong empirical support for our assumption that posterior samples should not be shared across decoding steps, validating the design choice in Assumption 3.1.
larger improvements
E.6 CASE STUDY
In this section, we present several representative examples from the MATH500 and GSM8K datasets, along with their corresponding solutions generated by Llama 3 2- Instrutt. We estimate token-level uncertainty for each output using the definitions provided Equ. 10-Eqn. 12. The visualizations shown in Fig. 7-Fig. 10, where Aleatoric Uncertainty (AU, in RED and Epistemic
Uncertainty (EU, GREEN) are visualized text-heatmap. The background Shading of each token corresponds to the magnitude of its uncertainty: the darker the shade, the higher the uncertainty, indicating lower model confidence for that token.
We observe that incorrect solutions consistently exhibit elevated uncertainty at or near the token where the wrong final answer is generated. For instance, shown in Fig. 7, sharp spikes in uncertainties happens with the arithmetic error of reversing "9600) 7200" into "7200 9600". In contrast, correct solutions tend show lower uncertainty overall and maintain low uncertainty key answer tokens.
Furthermore, incorrect outputs tend to contain a higher density of high-uncertainty tokens throughout the solution, whereas correct outputs generally more consistent and confident. These observations suggest that our token-level uncertainty estimation method can serve as a useful signal for identifying potential reasoning failures unreliable outputs, offering valuable diagnostic tool for both model interpretability and downstream error detection.
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Published as a conference paper at ICLR 2026
Problem Oula and Tona work for delivery service company, supplying different parts their state with milk products. They are paid $100 for each delivery made in a day. In particular month, Oula made 96 deliveries while Tona made 3/4 times as many deliveries as Oula. Calculate their difference in pay that month.
Ad
Stop
ato
tho number of deliveries made by
Stop
Calculate
uls
botal
pay
Lia
mado
onn
ona
made
Times
mary
deliveries as G 96 deliveriie ies and Ihey gei paid
$
100
each
de
Delivery
Bnd
Since
Lla
made
98 diaivvrr les made therr
be 95
100
Step
72 Iveries Slep Calo.ulat Calculate poid
pay
delierry
many
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pay
notai pay
be
AU
720 Stop Find the
offeremcc
ther pay
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delivery The difference in their pay 720
980
240
onn padd
and since there sPe
deliveries
The finzi anowor Is baxod 240
ona pay for the month
100
$ 720
Step Calcurate the citference between
O ua and
ona
Tma diference
960
720
final
Step
Calcurate number deliveries made
Step Calculate ukk
I0tal
made
made
deliveries
pay gM
they
delivery
Stop
horries
Caloulate Calculate ona total pay ona made
pay
wach
dellvary
paid times as many dalverias un which
100 and sinoo Ihere are
96 delivries uta pay 72 deliveries So their votal will be 72 100
EU
for the morth in 96 100
96n AB Step 3 720 *P Step Find The oifference
thei
Calculate the 10ta for
For each delivery
diference pay
960
since thare deliveries
answer
morrh
cifference
720
bosed
Correct
Incorrect X
Figure Case Study (1/4): The sample is from GSM8K, whose correct answer is 2400. In the incorrect solution, the model demonstrated significant uncertainty when mistakenly reversing "9600 7200" "7200 9600", and also exhibited high uncertainty the negative sign "_" in the final answer.
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Problem Kiarra twice as old as Bea. Job is 3 times older than Bea. Figaro years older than Job. Harry is half old Figaro. If Kiarra is 30, how old is Harry?
Stap
Dutormina
Eos
Ape
Bince
K
lar
ra Is
pki Stap
Datermine
Boa
AGR
based
on
lar
Ioloel
old
Bos
64.
50 yeurss
old
Bax
age
Since
K
30
Twico
old
Age
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byy dividing
inr
16
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by
2
Bea
Bea
Agg
ccn
CD culstedd
30
15
Be
30
Stop
2
Calcurane
JOb
age
bacod
B62
15
Step
MPO
Sinon
Fig
Job Smaw
older
Pan
Job
years
older Than Job and LOO
times order
than
be calouanted based
15
Step
AU
Bec
110
age
can
calculated
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100
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age
Bee
15
45
hatt
Fg
aro
Aig
Job
age
52 calculated
Final
AM
Step
Colcuuato
Harry age Sinio Harry hall
35
old
Fig
Harry
can Tound bty atviding
Fig
Harry
age
Fig age
02
26 The mhat
answer towed
26
020
AV Step
Determine
Ban
broed
K
Invical
years
Since
ckd
age con
by dhviding
ago
Ben
a00
ago
Carculate
Step
age
Job
Thonotore
yeare
calculated
EU
Determinc Harry
based
age
hall Therefore
caloulated
hal based
final
boxed
28
Correct
Incorrect X
Figure 8: Case Study (2/4): The sample is from GSM8K. In this example, the incorrect solution ignores the critical condition that "Figaro is 7 years older than Job," leading to the use of 45 instead of 52 the final calculation. Notably, the model exhibits high uncertainty the token "45" indicating lack of confidence in its own response at that point.
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Problem Simplify the following expression in $x$: $$19x 4x 81.$$
Fd
Step
Combince
like
torms
Combine
dtr
Stop
Comblic
like
terms
simpiiy
the
given
Simpl
Implty
the
ecconsssion
nead
combine
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Wil
performingg
(aziditions
right
15
90
The
1inal answer
boxed
Stop
AU
80
from
01
Step
Commone kike
terms
19
This ietvee
with
15
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expression boxed
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Step Combine like terms simpity the given
Step
Simpll
implty
by
expression we need combine
tho 1ho terma with The
performing
adeitions
right variable Jx Dombine and
Step
19
EU
15
80
The final
ArIswar
baxed
15
Stop Step
80
1$
from Combine Bie ferme us with Ad Step
leawes
Therefore the simplilied
eegreasion bosed 15
Incorrect X
Correct
Figure 9: Case Study (3/4): The sample is from MATH500, In this example, the incorrect solution gives its final answer "15x" in step 4. The model exhibits high uncertainty at the token next to "15x" because it overlooks the constant term. Furthermore, it can be observed that tokens associated with high uncertainty oceur frequently in the incorrect solution.
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Published as a conference paper at ICLR 2026
Problem In regular pentagon $FGHIJ$ extending the sides of the pentagon, as shown, forms star. What is the measure of angle $A$ in the figure?\n's [IMAGE)
Step
glvent
reggube
pert
Highet
Slep
given
regular
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agon
FG
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agon
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STAM
$
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ARR
where
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of the Orde
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angle
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101 usiize the
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that
the interior
polygon
part
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180
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polygcon
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apon
has
10B
1108
640
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subbti
uting
formul
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180
degrees
Step
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ating
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105
ciro
wnch
180
degroee
gives
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degreee
**
Step
SPM
Figur
given
that
angle
angle
imteroo
analcs
OT the
pont
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from
thc
total Sum oT angles
1:08
dagrees
and
angle
GH
108 dogrees
AG
Intwral
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sum of snglen
Step
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angles
linear
quidd
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300
and
have
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180 degreea
angle
100
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ongle
cam
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108
dogrees
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measure
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102
108
dic
108
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and wottendnng 1he wihwe
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whore 6 AC the rodius cirde
intertor angies an alto equas en Slep
lind angle
contoned facl that the sum the interiar ang e5 of any polygon can be caloulesed by number sides Of the polygon se Step
Stop
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the reguter peent agon
reguiar pert agan hrees
seglen
that adc up to froe 640
ciro
108
cir
180 degrees Where the
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8
180
dic
108
108
Since the
chr 108 ehc
108
dic
108
Gm
540
regular pent has sides WD can find the sum interirr
ciro which 1s true
Stop
New
We
nend
to
subbit uting
the lormun 9:
EU
wild up saglen the vertioes:
Since
190
Calcul
interior angla pamt agon 108 messune anglu
Find
5AD degrees Step degrees gives
Interir angles the pent agon ing sum
by autrrecl
The figure given shows that The angle 8 anglo
sum angice
108 dagreas
angle
degrees
QUadY Stop The angles
Step These angles form linear pair and add up to
angles 180
degrees Allowing us to mine that $ angle A PG
150
100
105 [6 Step 7 By subtract
216 degrees from
the measure angle A 150 The final
ligure
booed
Find the
by
pertor the subtraction
324
The tnall
boood
35
18
Correct
Incorrect X
Figure 10: Case Study (4/4): The sample is from MATH500. In this example, the model demonstrated notably high uncertainty at the incorrect answer token "60". In the correct solution on the left, the model had low uncertainty for the correct answer "36".
33


Reviews:
[{"review_id": "itCzz4U2Rg", "reviewer": "Reviewer_9t1g", "summary": "The paper introduces TokUR, a framework that lets LLMs self-assess and self-improve on multi-step mathematical reasoning by estimating token-level uncertainty during decoding. TokUR applies low-rank random weight perturbations, producing predictive distributions for each generated token. These per-token signals are decomposed into aleatoric and epistemic uncertainty and then aggregated to the sequence level to capture a response’s semantic confidence. Compared to prior query-level methods and response-level methods, TokUR offers a more fine-grained and accurate uncertainty estiomation. Experiments show that TokUR’s uncertainty is strongly correlated with correctness and robustness, flags incorrect reasoning paths, selects higher-quality solutions among candidates.", "questions": "1. It would be helpful to analyze the length-robustness of your token-level uncertainty estimates. As responses grow longer, how do the estimation errors (or bounds) evolve?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. Strong theoretical ground. \n\nThe paper’s theoretical backgrounds appear solid and thoughtfully developed. Assumptions are stated clearly.\n\n2. Clear positioning vs. prior work. \n\nThe manuscript does an excellent job for introducing the limitations of prior methods (query-level and response-level approaches), making it easy to understand the gap this work fills. \n\n3. Readable and well organized. \n\nThe paper is clearly written and easy to follow, with clear notations.", "weaknesses": "1. On task scope\n\nIt seems that the framework is “not limited to mathematical reasoning”. Is there a reason that the paper should focus especially on mathematical reasoning? A few non-math settings would support the generalizability of the framework.\n\n2. On cost analysis\n\nWhile the cost is the common challenge of uncertainty estimation, it seems that there is no thorough cost analysis in the paper, compared to prior methods like query-level uncertainty estimation. Also, how does the estimation technique used in Section 3.3 help reduce cost? \n\n3. On model diversity\n\nEvaluating two models from the same Llama family seems insufficient to support the correlation between accuracy and token-level uncertainty estimation. Including a few more models from different family and sizes is encouraged to reduce bias in experiments and to strengthen the claim.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "YFXOBZPS7x", "reviewer": "Reviewer_ZNDU", "summary": "This paper introduces the TokUR framework for uncertainty quantification in large language models (LLMs), targeting complex, multi-step mathematical reasoning tasks. TokUR estimates uncertainty at the token level by introducing low-rank random weight perturbations during decoding, enabling principled decompositions into aleatoric and epistemic uncertainty for each token and entire responses.", "questions": "- How does TokUR perform on open-domain, verbal, or dialogue tasks?\n\n- How much cost does in incur to get perturbed weights?\n\n- How do these perturbed weights affect the performance of the model?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- Most technical concepts are well-explained and visualized, aiding algorithmic understanding.\n\n- The approach itself is elegant.\n\n- It addresses a key practical need for safe and reliable LLM deployment, especially where confidence calibration is critical.", "weaknesses": "- Experiments focus on mathematical reasoning; application to verbal reasoning, open-ended tasks, or black-box LLMs is not shown.\n\n- Repeated sampling during inference, while compatible with vLLM, is still non-trivial and could limit deployment in latency-sensitive use cases.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "epgmjrtAtV", "reviewer": "Reviewer_YhsK", "summary": "This paper proposes the TokUR method, which injects low-rank weights to approximate the weight posterior distribution in Bayesian modeling of LLMs. During the decoding stage, it captures uncertainty in the model, leveraging quantized uncertainty to simultaneously estimate the semantic uncertainty of generated responses. Experimental results demonstrate that the proposed method produces uncertainty estimates that correlate well with task difficulty, and further show its effectiveness in detecting erroneous reasoning paths and improving answer quality.", "questions": "See weaknesses above.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "1. This paper introduces a novel approach for estimating uncertainty in long-form reasoning and demonstrates its effectiveness.\n2. The paper is clearly written and easy to follow.\n3. The experimental design is fairly comprehensive.", "weaknesses": "1. The authors decompose uncertainty into Aleatoric Uncertainty (AU) and Epistemic Uncertainty (EU). However, the definition of AU appears to correspond more to uncertainty in the input, while EU is more related to uncertainty in the model output. Since EU focuses on the model’s prediction behavior, it should theoretically better reflect the model’s capability issues. Nevertheless, the experimental results do not support this, and the paper does not provide analysis or interpretation of the three uncertainty estimates. I believe such analysis should be included.\n\n2. The expressions for Query-Level Uncertainty and Response-Level Uncertainty both involve (y), and the definitions appear highly similar, which may cause confusion or misunderstanding. I suggest the authors explicitly clarify the difference and explain the rationale behind these definitions to avoid ambiguity for readers.\n\n3. Assumption 3.1 ignores temporal correlations among parameters. The impact of this approximation needs to be justified either qualitatively or quantitatively to demonstrate the validity of the assumption.\n\n4. Some important related work is missing in the discussion. Many existing studies on uncertainty estimation and LLM reasoning theory are not referenced. For example:\n    *  *Position: Uncertainty Quantification Needs Reassessment for Large Language Model Agents* (ICML 2025), which discusses the relevance of uncertainty estimation.\n    * *A Theoretical Study on Bridging Internal Probability and Self-Consistency for LLM Reasoning* (NeurIPS 2025), which provides theoretical insights into reasoning errors in LLMs to improve reasoning accuracy.\n    * *From Calibration to Collaboration: LLM Uncertainty Quantification Should Be More Human-Centered* (arXiv:2506.07461), which analyzes the limitations of existing LLM uncertainty estimation approaches.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)