Martingale Score: An Unsupervised Metric for Bayesian Rationality in LLM Reasoning
Authors: Zhonghao He
Tianyi Qiu
Hirokazu Shirado
Maarten Sap
Venue: NeurIPS 2025 poster ()
Keywords: Martingale Score
Martingale property
bayesian reasoning
LLM
unsupervised
evaluation
belief entrenchment
forecasting
Primary area: evaluation
TL;DR: We introduce the Martingale Score, an unsupervised metric from Bayesian statistics, to show that reasoning in LLMs often leads to belief entrenchment rather than truth-seeking, and shows this score predicts ground-truth accuracy.

Abstract: Recent advances in reasoning techniques have substantially improved the performance of large language models (LLMs), raising expectations for their ability to provide accurate, truthful, and reliable information. However, emerging evidence suggests that iterative reasoning may foster belief entrenchment, rather than enhancing truth-seeking behavior. In this study, we propose a systematic evaluation framework for *belief entrenchment* in LLM reasoning by leveraging the Martingale property from Bayesian statistics. This property implies that, under rational belief updating, the expected value of future beliefs should remain equal to the current belief, i.e., belief updates cannot be predicted from solely the current belief. We propose the unsupervised, regression-based *Martingale Score* to measure violations of this property, signaling a deviation from the Bayesian ability of updating on new evidence. In open-ended problem domains, including event forecasting, value-laden questions, and academic paper review, we found such violations to be widespread across models, reasoning paradigms, problem domains, and system prompts, where the future beliefs are consistently predictable from the model's current belief, a phenomenon which we term *belief entrenchment*. Through comprehensive experiments, we identify the models (e.g., GPT-4o), reasoning techniques (e.g., chain of thought), and domains (e.g., forecasting) more prone to belief entrenchment. Finally, we validate the Martingale Score by showing that it predicts ground-truth accuracy on problem domains where ground truth labels are available. This indicates that, while designed as an unsupervised metric that operates even in domains without access to ground truth, the Martingale Score is a useful proxy of the truth-seeking ability of the LLM reasoning process.

Martingale Score: An Unsupervised Metric for Bayesian Rationality in LLM Reasoning
Zhonghao He* University of Cambridge 2h3780cam. ac. uk
Tianyi Qiu" Peking University qiutianyi qty0gmail,... com
Hirokazu Shirado Carnegie Mellon University shirado8cmu. edu
Maarten Sap Carnegie Mellon University maartensap@cmu, edu
Abstract
Recent advances in reasoning techniques have substantially improved the perfor- mance of large language models (LLMss), raising expectations for their ability provide accurate, truthful, and reliable information. However, emerging evidence suggests that iterative reasoning may foster belief entrenchment, rather than en- hancing truth-secking behavior. In this study, we propose systematic evaluation framework for belief entrenchment in LLM reasoning by leveraging the Martingale property from Bayesian statistics. This property implies that, under rational belief updating, the expected value of future beliefs should remain equal the current belief, i.e., belief updates cannot be predicted from solely the current belief. We propose the unsupervised, regression-based Martingale Score measure viola- tions of this property, signaling a deviation from the Bayesian ability of updating on new evidence. open-ended problem domains, including event forecasting, value-laden questions, and academic paper review, found such violations to be widespread across models, reasoning paradigms, problem domains, and system prompts, where the future beliefs are consistently predictable from the model's 's current belief, a phenomenon which we term belief entrenchment. Through compre- hensive experiments, (e.g., chain of thought), and domains (e.g.. forecasting) more prone to belief en- trenchment. Finally, validate the Martingale Score by showing that it predicts ground-truth accuracy on problem domains where ground truth labels are available. This indicates that, while designed as unsupervised metric that operates domains without access to ground truth, the Martingale Score is useful proxy of the rruth-seeking ability of the LLM reasoning process.
identify the models (e.g.. GPT-44),, reasoning techniques
in
Introduction
Consider a stubborn person who refuses to be shown wrong, or king surrounded by syco- phants. Human beings, their efforts seek true beliefs. often end up entrenching their pre- existing beliefs, whether due their own confir- mation bias [Klayman 1995) or due to external
Should the state of New York (NY) be split into two states?
Polls show limited enthusiasm. 30% Weak No
Spitting couses administralive headache 20%
Calffrnia repoated proposals have tailed 10% Strong No
Figure 1: Example of Belief Entrenchment: LLM progressively updates beliefs favor of prior belief. Its belief update is highly predictable from the prior, violating the Martingale property.
Eqqual contribution.
'These authors jointly supervised this work.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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confirmatory influence [Cinelli et al. 2021 Shi-
rado et al. 2020). , Research suggests that such an entrenchment of belief lies at the heart of social
epistemic problems such as polarization and misinformation [Del Vicario et al. 2017 Modgil et al.
2024| Lefebvre et al. [024]] Calhoun 2004), and may be responsible for a wide range of downstream
cognitive biases [Oeberst and Imhoff 2023 The entrenchment happens in the information processing
of humans, a.k.a. their reasoning [Oeberst and Imhofi 2023).
Having learned patterns of human reasoning from human-generated text data at an Internet scale, large language models (LLMs) are designed help humans by answering their questions [Luo et 2022] or assisting with their tasks (Zheng et al. 2023 Following their initial success in sophisticated
cognitive tasks [Brown al.) 2020 Zhao al. 2023]. range of reasoning techniques have emerged
based on these models. From the earliest Chain-of- ohannof-Thooght (CoT) technique (Wei et al., [222] to the more recent paradigm of inference-time scaling via reinforcement learning /Zllikaan et al.. 2024 Jaech et al., 2024 Guo et al., (025) (reinforced reasoning henceforth), these methods aim to help language models in truth-seeking, i.e., gaining a correct understanding via argumentation, evidence-sceking, and trial-and-error at inference time.
However, such reasoning in language models can deviate from truth-seeking due to belief entrench- ment -a systematic tendency to update beliefs in favor of prior opinions rather than in response to new evidence (Figure D.. When this occurs, it can degrade the accuracy of model's Is conclusions mirroring well-documented effects in human cognition (Park et al. 2010] and mislead users with an unjustified sense of confidence (Shi et al.) 2024 Zhou al. 2024a Although recent studies suggest the presence of belief entrenchment in L.MMs Schmidgall et al. 2024 Shi et al., 2024 Sumita al., 2024), demonstrating it rigorously remains challenging. any single example, is difficult to distinguish a justified, prior-consistent update (e.g., a prediction later validated by evidence) from biased update (c.g., representing evidence to maintain an unfalsifiable prior) [Atallah et al.) 2202 Ji. 2023]. As a result, prior work relies on highly synthetic tasks or domain-specific setups where belief entrenchment is easy to detect [Schmidgall et al. 2024]. However, these methods fall short in assessing reasoning failures as they unfold, especially in open-ended tasks where the ground truth is ambiguous or unavailable.
To address this challenge, we propose a statistical measure of a reasoning model's 's tendency toward belief entrenchment. Specifically, we define belief entrenchment as violation of the Martingale property core principle in Bayesian reasoning which, informally, states that the direction of belief updates should not be predictable in advance [Chamley, 2004 Molavi 2021]. This motivates using the Martingale violation principled and unsupervised indicator of irrational belief entrenchment. Concretely, if model's future belief can be reliably predicted from its prior belief across reasoning verations-qqaniiin via the goodness-of-ff fit of a regression model- this indicates a deviation from the Martingale property and thus the presence of belief entrenchment.
We empirically validate our measure's usefulness through experiments on three domains: event fore- casting. value-laden questions. and academic paper review (hereafter, Forecasting, r/ChangeMy View, and OpenReview, respectively). Across a wide range of models (e.g., GPT-40,, I.aaa 4), reasoning techniques (e.g., COT, Debate), and system prompts, we find that belief entrenchment is a pervasive phenomenon. Our results show that future belief updates are consistently predictable from prior beliefs, clear violation of the Martingale property. Critically, we validate the Martingale Score by demonstrating that it strongly correlates with a drop in ground-truth accuracy (measured by the Brier Score) in domains where such labels are available.
Contributions
The Belief Entrenchment Problem and the Martingale Score. We define belief entrenchment, a statistical property that quantifies confirmation bias in LLM reasoning. We then introduce the Martingale Score, the predictability of belief updates based solely on prior, as unsupervised and domain-agnostic measure of belief entrenchment.
Uncovering Widespread Belief Entrenchment in LLM Reasoning. We use the Martingale Score to evaluate mainstream LLMs and find that belief entrenchment is a pervasive phenomenon across different domains (Forecasting, r'Change and OpenReview), prompts (prior- conforming, no-prompt, critical-thinking). and model families (GPT, DeepSeek, Gemini, Llama, etc).
2

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Connecting Belief Entrenchment to Accuracy Loss. We find that belief entrenchment con- sistently predicts an accuracy drop in problem domains where ground truth labels exist (e.g.. forecasting). This suggests that the Martingale Score serves proxy for reasoning quality, even in settings where the ground truth is unavailable or still unfolding.
2 Related Work
Bayesian Rationality in Language Models In the LLM context, to evaluate Bayesian rationality is to evaluate the capacity to incorporate evidence via in-context learning in a way that is consistent with Bayes's 's theorem. Gupta al. (2025) presents that LLMs could follow Bayesian update when given a large amount of coin flips, despite a biased prior. However, more negative results appear in realistic settings with complex natural-language features: [Qiu et al. 2025a| demonstrated that LLMs do not update their belief's of user preferences in multi-round interactions as expected by the Bayesian framework. [Falck et al.. 2024 falsifies the hypothesis that LLM in-context learning is Bayesian. (Zhao et al. 2021 Wang et al. 2023) demonstrated that LLMs are sensitive to the arrangements of examples prompts. Solutions are proposed to make the process of in-context learning or reasoning more Bayesian: mimicking the predictions of an ideal Bayesian reasoner [Qiu et al. [025a]] abstract reasoning (Zhou et al., 2024b], and combining abduction and deduction [Feng et al.) 2024 However,
they tend to require auxiliary structures such as Bayesian networks, limiting their practical use.
Cognitive Biases in Language Models and in Humans LLMs suffer from a variety of cognitive biases they trained large corpus of human data (Echterhoff et al., [024]. They also appear to employ biases distinct from those of humans when being deployed as judges, such as position bias (favoring certain positions). verbosity bias (favoring long answers), sentiment bias (preferring
positive expressions) Yee et al. 2024). and certainty bias (hhu et al. 2024a). Among all the cognitive biases, confirmation bias is of our particular attention as it is believed to be the root of polarization
(Atallah et al., (022]. It is also hypothesized that most cognitive biases are variants of confirmation bias [Oeberst and Imhoff 20023. Given these, we focus our attention on confirmation bias and use belief entrenchment to operationalize it in the LLM context.
Truthful AI and Truth-seeking AI Research in Truthful AI focuses on developing Al systems that output truth of the real world [Lin et al.) 2021]. Previous work proposed interventions to improve "truthfulness" such as discovering learned activations aligned with truth [Burns et al. [2022] or shifting model activations during inference time (Li al.) 2023). and trade-off between truthfulness and utility (Su et al., 2025). Honest Al emphasizes that Al does not "intentionally" assert anything that does not align with its beliefs in pursuit of rewards [Evans et al.
[221]] This is closely related to the
concept of deception in LLMS [Hubinger et al. 2024 Su et al. 20255. In contrast, we are interested in less investigated concept: truth-seeking Al Koralus 2025 the process of weighing evidence and discovering novel findings. seuth-seeking may become an important objective of LLMs as they becoming more agentic [Chan et al., 2023), where the decision of what further information to seek is dependent on how the LLM interprets the current situation. Truth-seeking Al explicitly aims for gaining truth, but "truth-seeking" may not be single metric optimize against. Being Bayesian could be one target, and coherence might be another [Wen et al. 2025).
3 The Problems of Belief Entrenchment
3.1 Definitions
Belief' entrenchment is the systematic tendency update one's beliefs in favor of one's existing leanings rather than against, regardless of evidence. It is closely related to confirmation bias in cognitive science, where agents proactively seek, interpret, or recall evidence that is in favor of existing beliefs [Nickerson] 1998). Confirmation bias is concerned with information processing at the individual instance level, and measuring it is difficult. In psychology, the measure of confirmation bias is task-specific and of low reliability. More importantly, to measure confirmation bias, the tendency of interpreting information that supports one's beliefs and values, requires reliable access to This is significant challenge by itself.
and measure of one's 's prior beliefs [Berthet 202
3

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In light of the difficulties, we use the statistical violation of the Martingale property from Bayesian statistics as the empirical measurement of belief entrenchment. Martingale property states that the expectation of the posterior, conditional on priors, should always equal the expectation of prior [Molavi] 202T]. other words, under the Martingale property of Bayesian statisties, reasoners should never make predictable belief updates. Different from confirmation bias, because of how belief entrenchment is measured, is defined as a statistical property, rather than an individual tendency.
The term "beliefs" is used loosely in this study. We do not discuss questions such as whether LLMs could hold beliefs the way humans do. Rather, what we are concerned with LLMs' expressed "beliefs" through their output when LLMs are entrenched by their own "beliefs," they express high confidence their own stated output, unjustified sense of confidence would mislead LLM users.
Belief
Belief update Relatedly, belief update refers to the change in confidence in LLM's s output. And we use LLM judge to assess confidence, as detailed in Section [5]] We treat the process of having LLMs do chain-of-thought reasoning belief-updating process, similar to having a human being extensively reason about certain problems. In this process, the very beginning of model output is treated as "prior belief",, whereas the very end of model output, after extensive reasoning or engagement with external evidence, treated as "posterior belief"".
3.2 Risks Brought on by Belief Entrenchment
We identify three levels at which belief entrenchment poses risks: the model's reasoning capability, reasoning evaluation, and human-Al interactions.
Bayesian Reasoning and Truth-Secking Previous research presents mixed results to the question whether LLM in in-context learning is Bayesian [Gupta et al.) | Falck et al. (202)). Meanwhile, specific reasoning failure instances were reported: sycophancy [Sharma et al., 2023]. J. inverse scaling [Gema et al. 2025]. following group majority rather than sticking to the truth [Weng et al. 2025
Those specific failure cases of Bayesian reasoning compromise LLM task performance.
Failures in Reasoning Evaluation Empirically, LLM reasoning improves performance on a variety
of tasks, including mathematics [Lu et al. 2023 and coding [Gu et 2024). Reasoning evaluation is usually conducted by measuring ground truth accuracy [Sawada et al. 2023 Phan et al. 2025). Such outcome-based based evaluation falls short of our expectations because does not tell us how LLM reasoning achieves superior performance, hence whether it would generalise -ff--iiirtriuttionn If it is achieved by recalling parametric beliefs acquired from pre-training, LLM reasoning would have limited utility in real-world tasks because, in real-world tasks, problem-sol need to take contextual information into account. We need process- based metrics answer the question of "how" [Mondorf and Plank 2024a]
Misleading Human-Al Interactions Research sycophancy (Oeberst and Imhoff [002]]. and conformity (Weng et al. (025) demonstrates that L.MSs may favor pleasing human users or following the group majority over truth. This may mislead users into entrenchment of their own fallacies or biases. On the other hand, is also known that human reasoning suffers from confirmation bias, and it causes downstream social epistemic problems, such as polarization [Lefebvre et al., 2024] and misinformation [Shirado et al. 2020). The feedback loops between humans and LLMs (that humans acquire beliefs from LLMs that trained on data containing human beliefs) may lead to lock-in of
false beliefs collectively [Burton et al. 2024 Qiu et al. 20255 Weidinger et al. 2023
4 Measuring Belief Entrenchment with Martingale Score
To address these problems, we propose the Martingale Score as principled, unsupervised measure of belief entrenchment in LLM reasoning. Effective reasoning requires the capacity to update beliefs in response to new evidence, property aligned with Bayesian rationality. The Martingale Score quantifies the degree to which belief updates can be predicted from prior beliefs alone; a higher score indicates stronger entrenchment and weaker responsiveness to new information (Figure 2).

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Predict
Predict
Predict
Predict
Prior Bellef
60%
Prior Bellef
60%
Prior Bellef
40%
Prior Bellef
20%
70%
60%
80%
70%
30%
0%
-30%
70%
80%
Bellef Update
Belief Update
20%
Bellef
60%
Bellef Update
Updato
Martingale Score calculates the statistics of predictability of belief update solely based on prior.
Figure 2: An illustration of Martingale Score calculation in our setting. "Prior Belief" refers to the expressed beliefs in most immediate LLM output; whereas "Posterior Belief" usually refers to the terminal beliefs after extended reasoning or engagements with external evidence. "Prior" and "Posterior" are relative concepts and their difference taken as "Belief Update". We estimate the linear coefficient when running linear regression between belief updates and prior beliefs. The positive value of the linear coefficient our practical choice of the Martingale Score, measuring the predictability of belief update solely based on prior belief.
4.1 Defining the Martingale Score
To compute the Martingale Score, we perform the regression Ab = B1 Upriorr + 3o + e, where Oprior are the prior probabilities, Ab = Uposterior Uprior, and is the error term.
We define the sample estimate $ of the linear coefficient the Martingale Score M, with the Ordinary Least Squares (OLS) method. Equivalently, when there are samples,
M = En (Ab, (b)rioor bprior
(I)
1 (bprior,t
bprior) )2
Martingale Score measures the extent to which the prior belief Oprior positively (or negatively, if M < 0) predicts belief update Ab. Using OLS allows to test the statistical significance of M, assessing whether the relationship between Ab and bptior is distinguishable from zero (e.g.. via a t-test with p 0.05).
We choose the M = 3 definition for its simplicity, lack of confounders (as opposed to R (coefficient of determination) of a logistic regression that introduces confounders such as the intrinsic variance of belief update not attributable to prior belief), and empirical reliability (as opposed to logistic regression on the binary direction update, which neglects the magnitude of belief updates and produces random- random-seeming seeming results).
4.2 Theoretical Justification for the Martingale Score
The Martingale property states that the expectation over one's posterior, conditional on their prior, should always be equal to the prior [Molavi 2021]. Formally,
E Ab bprior = p = 0, Vp E [0.
This implies that the direction of Bayesian agent's belief update (whether positive or negative) should not be predictable from the prior alone. Indeed, the Martingale property been shown be the defining characteristic of Bayesian rationality [Molavi 2021].
The Martingale property implies that the prior bprior statistically exogenous to the belief update 2011. This exogeneity yields two desirable properties: the expected coefficient B[ = 0 in the regression ( and consistency of the estimator, i.e., 8 - 0 in probability as the number of samples approaches + [Hayashi] 2011]. These properties justify the Martingale Score, defined in (I) principled measure of violations of the Martingale property (2). Formally,
Ab Hayashi

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deepsook
?
Claude
CharGPT
Should the state of New York (NY) be split into two states?
Gemini
Spitting might diutt Nitx cutture identity 30% Weak
Chain of Thought Reasoning
X LLaMA
Spliting might reduce Ni's political influence.
25% Weak No
Reasoning paradigms.
(d) Models.
show enthusiasm.
20% No
Forecasting
Reasoner
Judge
Spitting causes adminietraivee headsche.
15% No
No System Prompt
"Change My View"
Prior-conforming
Caltornii repeated propasala have failed.
10% Strong No
Critical Thinking
Chain of Thought Reasoning
Review Academic Papers
(e) System Prompts.
(a) An example of real reasoning trajectory evaluation pipeline.
(c) Domains.
Figure 3: All experiment setups. (a) Example reasoning trajectory. (b) CoT reasoning and debate reasoning, where in the latter, one model is instructed to debate with its clone. (c) Problem domains: forecasting questions from Metaculus and Polymarket; value-laden questions from r/ChangeMy View, where owners of posts share statements they hold strong beliefs in, expecting counterarguments: acceptance decisions for ICLR submissions given the abstract and reviews. (d) Models evaluated. (e) System prompts used. "Prior-conforming prompt" instructs the model to fixate on their prior beliefs, whereas "critical thinking" prompt encourages models challenge their prior beliefs. The two prompts represent the extreme behaviors we intend to demonstrate.
Proposition 1 If the Martingale property holds, the population coefficient 31 is 0 and the sample estimate 31 is an unbiased and consistent estimator of fr. In this specific case where 31 = 0, this implies E(M) = 0 and M & 0, as n + oo
As the Martingale property is prerequisite for Bayesian reasoning (i.c., ensuring that updates driven by new evidence and not by prior views), its violation indicates belief updates are systematically predictable from priors. And the violation of Martingale property, in turn, defines the core of belief entrenchment. We thus operationalize belief entrenchment as the degree to which a model violates the Martingale property.
Experiment Setups
We set up experiments evaluate belief entrenchment with the Martingale Score for a broad coverage of tasks (detailed in Section 5.2 and how it affects accuracy in domains where there is ground truth. The implementation details can be seen in Appendix C
5.1 Using LLM Judge to Measure Beliefs
LLMss are often poorly calibrated: their confidence scores-wheite expressed through token proba- bility or self-reported scores do not reliably reflect their true degree of belief [Paiitan and Holmes 2024 Zhou et al., 2023). To address this, we adopt an "expressed belief" approach rather than relying on "internal beliet." which remains an open research problem (ase et al. 2021 Scherrer et al.) 2023 Herrmann and Levinstein| 2024). Expressed beliefs are inferred from the model's outputs and are more consistent with how users experience and interpret LLM responses in practice [Zhou et al.) 2024a] To extract these beliefs, we employ a separate "judge" model (e.g., GPT-4o) that assesses each model's reasoning steps and assigns belief score E [0, 1]. To ensure robustness, evaluate
Refer to Appendix A for for
the proof.

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multiple judge models and confirm that the results are consistent across them. Details can be seen in Section 6.2
5.2 Problem Domains
To study belief entrenchment in LLM during the reasoning process- we select domains that meet the following criteria:
how models incorporate new evidence
Not solvable by memorization. The domain should include questions that cannot be answered using information seen during pretraining. For example, we target events or facts that were resolved after the model's is knowledge cut-off.. If a model was trained up to August 2024, it cannot know who won the 2024 U.S. presidential election without access to external tools.
Contain new evidence that could shift beliefs. The domain must include incoming evidence that Bayesian reasoner would use to revise its beliefs. This allows to evaluate whether LLMs appropriately update their beliefs in response to new information, or whether they remain anchored to prior assumptions.
Ground truth becomes available after models' knowledge cut-off. The domain should provide verifiable ground truth labels after the model's knowledge cut-off date. This enables us assess whether belief entrenchment correlates with a drop in final accuracy when models fail to adapt to post-cut-off evidence.
Based on these criteria, we choose three domains for evaluating belief entrenchment: forecasting, value-laden questions, and academic paper review.
5.2.1 Forecasting
We source forecasting questions from Metaculus [Metaculus] 2015] and Polymarket [Polymarket 2020] to test belief entrenchment in LLMs. We choose this domain for two key reasons: (I) forecasting questions come with ground truth labels once resolved; and (2) achieving high accuracy of forecasting questions reflects Bayesian-like reasoning, it requires seeking evidence and proportionally updating beliefs. While forecasting differs from factual question answering, accurate forecasting nonetheless requires well-calibrated, unbiased belief updates. This makes it strong proxy for rational reasoning under uncertainty.
5.2.2 Value-Laden Questions
To assess belief entrenchment in subjective or controversial domains, we use questions from the r/ChageeMy View subreddit [Tan af., 2016). These discussions are explicitly designed explore whether individuals (or in this case, LLMs) can revise their opinions when presented with counter- arguments. This allows us to examine whether LLMs update their value-oriented stances during multi-step reasoning or remain anchored to prior views.
5.2.3 Academic Paper Review
Scientific peer review is another setting that satisfies all three criteria for evaluating belief entrench- ment. We use the open-access ICLR submission dataset from OpenReview (Höpner et al.. 2025). which includes paper abstracts, bibliographies, reviewer comments, rebuttals, and final acceptance decisions. In our setup, the model is prompted to act as an area chair, making final acceptance decision based on the abstract and the arguments presented in the reviews and rebuttals. This task allows us to evaluate how reasoning unfolds when prior impressions (e.g., from the abstract) may conflict with later-stage evidence (e.g., critiques and responses).
5.3 Reasoning Techniques
evaluate belief entrenchment with two reasoning techniques: Chain-of-Thought (CoT) Wei et al.. 2022 and debate [Khan et al. 2024). Debate lets two clones of the same expert model hold opposing positions on topic and participate debate, allowing less informed observer arrive at better understanding of the subject.
7

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Table 1: Martingale Scores under different setups. CT is short for critical thinking, while PC is short positive Martingale Score M indicates that per unit increase in Opriors there is an M-unit increase in Ab. Entries in this table measures belief entrenchment per reasoning step, and the bias may add up to much high levels during the full reasoning trajectory. Martingale Scores whose t-test produces p < 0.05 are marked with"" "
for prior-conforming.
Forecasting
ChangeMy View
OpenReview
Col
Delote
CoT
Debote
Col
Debate
GPT-4o (May 7)
Prowpp
10.0018
10.0671*
10.0734'
Prompt
+0.0156
+0.0859"
Prompt
-0,0227
+0.1455"
Prompt
+0,0207
+0,0366
DeepSeek
Pronept
+0.0511
+0.0305"
+0.18607
Pronept
+0.0450"
+0.0487
+0.0526*
+0.0961
+0.0689
+0,0299
Pronept
+0.0335
0.0929
+0.1155
+0.0739
+0.1025*
+0.1337
DeepSock
Pronapt
+0.0316
-0.0064
+0.0890"
+0.0179
+0.0805"
+0.0713
Pronept
PC
+0.0763
-000216
+0.0579
+0.0511
-0. -0.1495"
+0.2113'
Prompt
+0.0764
-0.0196
+0.1209"
+0.0900
+0.1012
+0.0882
Gemini Flash
CT Pronupt
+0.0067
0.0012
+0.11 10.1"
+0.0642
+0.0817
+0.1263
Promppt
PC
1 10.0335
0.0368
1..1022
0.0296
0.0849
0.0044
Prompt
No
1 1.00500
00,0088
0.14200
0.0900
1..0000"
0..116
Liama Scout
CT Prompt
+0.0125
0.0395
+0.11496
+0.0233
+0. +021
+0.11 40.1729
Prompt
PC
40.0740"
0.0114
+0.1372
+-0.0003
0.0253
+-0.1 1029
Prompt CT Prompt PC
No
#0.0178
+0.01003
+0.10038
+-0.1100"
+000823*
+0-17499
Llama Maverick
I 10,0282*
10.0132
10.1161
0.1188
1..0000
D.22511
Promptt
+0.0523
00.0128
+0.14355
+0.1600
+0.0951"
+0.1724"
5.4 Models and Prompts
We conduct experiments using GPT-4o, DeepSeek R1, DeepSeek V3, Gemini 2.0 Flash, LLaMA 4 Scout, and LLaMA 4 Maverick. To examine how prior beliefs affect reasoning, we manipulate model priors via system prompts. Specifically, in addition to a baseline with no system prompt, we introduce two prompting conditions: prior-conforming prompt and crificallthinking prompt.
The prior-conforming prompt reinforces belief aligned with the model's likely initial stance, serving both as sense check and as a potential lower bound for belief entrenchment and accuracy. In contrast, the critical-Be critical-thinking thinking prompt encourages openness to counter-evidence and rational belief revision. All prompts used in the experiments are provided in Appendix C
6 Results
This section presents empirical findings. We first establish association between belief entrench- ment (as measured by the Martingale Score) and drops in ground-truth accuracy, as a vindication for the former. We then benchmark range of closed- and open-weights models different problem domains and setups, and causally attribute belief entrenchment to various factors like models, prompts, and reasoning techniques. Finally, we confirm LLM-as-a-judge validity.
6.1 Belief Entrenchment Results
Belief Entrenchment is Prevalent Across Setups Table the Martingale Score of different models under different experiment setups. A positive Martingale Score M indicates that there an M-unit increase in Ab per unit increase in lprior- In most of the experiments, including almost all of those with CoT (51 out of 54), we see positive Martingale Scores, suggesting consistent belief entrenchment.

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CoT en Faracisting
Forecasting
CoT on OpenReview
Debaie OpenReview
Brier Score (Lower is Better)
0.1
0.0
Martingale Score (Bellef Entrenchment: Lower is Better)
Figure 4: Relationship between the absolute value of the Martingale Score and the Brier Score. The former indicates the predictability of belief update based solely on the prior belief, and the latter measures the accuracy of probabilistic predictions. We observed that the Martingale Score and Brier score are positively correlated across all setups, suggesting belief entrenchment harms accuracy on binary problems. Taking "CoT on Forecasting" as example, Martingale Score of 0.0 corresponds to Brier score smaller than 0.25, slightly better than random guess (0.250) [Halawi et al. 2024]; in contrast, when the model is mildly entrenched on its prior belief (marked by Martingale Score of 0.04), the forecasting performance is worse than random guess.
Notably, we observe overall more severe belief entrenchment in value-laden domains such r/Change- MyView (when using the same model, under the same system prompt, the Martingale Score is always higher in r/ChangeMy View than in Forecasting), suggesting that belief entrenchment might be a more serious concern domains where we cannot evaluate LLM performance against ground truth. We also intended manipulate the degree of belief entrenchment by using critical thinking prompt and prior-cone coffor-informing prompt. We observe that belief entrenchment is more severe under a prior-conforming prompt and less severe under critical thinking prompt in Forecasting, but not in rCChangMyy View or OpenReview. More discussion about factors influencing belief entrenchment can be seen in Appendix B
Belief Entrenchment is Not an Artifact Under "prior-conforming prompt", belief entrenchment meant happen, LLMS instructed to emphasize arguments in favor of their prior be- lief, and such reasoning harms performance. However, we noticed that even under "no system prompt" and "critical thinking prompt", belief entrenchment also happens, although (MPrior-confominng 0.082 + 0.018, M No-propp 0.075 + 0.014, M Crirical-thinkngg 0.072 + 0.018, with 95% CI). The results are significant, demonstrating a consistent tendency of belief entrenchment in LLM behavior.
lesser extent
Belief Entrenchment Harms Accuracy As the ultimate aim reasoning LLMs is to obtain true beliefs, we show that belief entrenchment diminishes this objective. We do so within problem domains where ground truth labels are available (i.e., Forecasting and OpenReview).
Figure Nigure4shows showss the correlation between the absolute value of the Martingale Score (lower is better) and the Brier Score (measuring prediction accuracy by the mean squared error between a predicted probability and the actual outcome; lower is better). Each data point represents the Brier Score and Martingale Score one setup (of one model, with one type of system prompt and one reasoning mode, 100 questions in one problem domain.

---

Signfiicnce abs(Martnggcle) Gost 0.05 (Sigrificnt)
0.5
p=0.011"
P (Margially Significant)
p=0.016
Cuefficlent of abog/Martiggaee
0.4
(Not Signficcnt)
p=0.045
p-0.356
p-0.214
variance explained by Martingate)
0.40
2
R with abs(Martiggale)
AR1-0.051
abx(Mariingsle)
0.35
Model R-squared
0.80
ARt-0.007
0.20
ARP-0.038
0.15
AR 011
0.05
AR100.020
0.00
M2 (-Domain)
abss(Maringgee)
(-Domain
(+Domein +RM RRM Prompt)
Model Specification
Figure 5: Increased absolute value of the Martingale Score is associated with worse prediction accuracy (higher Brier Scores) and explains a significant portion of the latter's variance. In each regression model, we predict the Brier Score with the absolute value of the Martingale Score, while controlling for different potential confounders, including problem domain, reasoning techniques ("RM"), choice of model, and choice of prompt.
While outcome-based metrics (such Brier score) measure model performance, they reveal little about why models achieve high performance, and for this reason, we need process-based metrics reasoning [Mondorf and Plank 2024b). We show positive correlation between the Martingale Score and Brier Score in Forecasting, suggesting that belief entrenchment is one of the culprits that compromise LLM reasoning.
Martingale Score Can be Used in Open-ended Domains It's worth noting that the Martingale Score, being an unsupervised measure, is directly applicable in open-ended problem domains (e.g.. rChangeMy View) as well. Table Table demonstrates that, least with CoT, belief entrenchment consis- tently happens regardless of prompt types and models. The comparable level of belief entrenchment (McMveco = 0.103 + 0.013, M Forecasting CoT 0.037 + 0.011, M OpenReview CoT 0.086 + 0.012, with 95% CT) corresponds to accuracy drops forecasting and OpenReview (as in Figure Figure 5). where ground truth exists, suggesting consistently worsened judgment under belief entrenchment. Ruling out potential artifacts (presented in Section 6.11. we thus think the Martingale Score is valid metric of reasoning paradigms with the utility of understanding the quality of reasoning.
6.2 Judge Consistency Evaluation Results
In our cross-judge consistency and human-L..M agreement analysis, all judges (L.LMs or humans) show strong correlation with GPT-4o.
10

---

Cross-LLM Agreement We construct pairs of LLM judges (e.g.. GPT-4o VS Gemini-2.5 pro. GPT-4o VS DeepSeek-v3) and see how much their belief evaluation correlates with each other. Note that we've acquired GPT-4o data for all problems but significantly less with other judges (from 283 problems with GPT-4. 1-mini 3,844 with DeepSeek-v3), so we set up GPT-4o as the default judge for all comparisons, in line with our choice of judge in the main experiments.
Human-LLM Agreement We use a small batch of human evaluation data to validate the LLM judge (i.e., human-LLM consistency evaluation). Specifically, we request human evaluators to do belief evaluations exactly like what we request LLM judges to do; we then construct pairs of Buman-LLMy judges (c.g., human evaluator VS GPT-4o). Full results can be seen in Table
Table 2: Inter-rater Agreement of Judges with GPT-4o.
Rank
Judge Model
Batches
Problems
Belief Samples
Pearson
Spearman p
p-value
1
Human Evaluator
2
20
195
0.8822
0.8770
< 0.001
2
DeepSeck-v3
48
3,834
24,921
0.7774
0.7620
< 0.001
3
HPPT-41-minn
3
283
2,015
0.7581
0.7490
< <0.001
4
Gemini-2.5-po
4
373
1,688
0.7460
0.7230
< 0.001
5
Human Evaluator
2
18
173
0.7152
0.6812
< 0.001
All judges show a large positive correlation with GPT-4o, and all results statistically significant (p < 0.001)
Considering results from both cross-judge consistency analysis and aumaa-evaluation validation, it is highly unlikely that our belief entrenchment results stem from judge bias.
Conclusion
In this paper, we propose the Martingale Score principled and unsupervised measure of belief entrenchment LLM reasoning. We show, as validation, that the Martingale Score predicts the ground-truth accuracy of reasoning process, and then apply it to range of different models, prompts, and problem domains. We conduct analysis and identify a collection of factors that increase or alleviate the severity of belief entrenchment.
Limitations In OpenReview, this study does not demonstrate the correlation between Martingale Score entrenchenn)))))) and Brier Score (accuracy). We think this is because ground-truth labeling in domains such as OpenReview is community-voted (in this case, paper acceptance decisions), and comeunity-voted decisions can be noisy [Beygelzimer al., 2023). However, Table [ shows that belief entrenchment is more severe phenomenon in domains where human judgments are required (such as r/ChangeMy View and OpenReview). This raises concerns over adopting such LLMs in those domains.
Future Work Due to resource constraints, we have not systematically studied belief entrenchment in reinforced reasoning [uuo et al. 2025), despite the recent popularity of this approach to LLM reasoning. Besides, the violation of ideal Bayesian rationality can be demonstrated by both irrational engagement with internal reasoning process and external evidence. In this study we only focused on the former. Future study can include an "evidence searching" component to evaluate LLM's capacity to update belief in response to new external evidence (as opposed to its propensity to fixate on prior belief). The difficulty here is experimenters need guarantee that the evidence searching does not give LLMs under evaluation direct access to ground truth, if such ground truth exists. In domains with ground truth (e.g., Forecasting, but not OpenReview). we demonstrated the causality of belief' entrenchment and accuracy. Future research can test the validity of the Martingale Score in open- open-ended ended domains by demonstrating other downstream consequences of belief entrenchment.
To demonstrate the usefulness of Martingale Score as ceneral-purpost process-based reasoning evaluation, one could extend our pipeline into the evaluation of sycophancy [Sharma al. 2003).
*As a reference point on a different setup, the NeurIPS 2021 review consistency experiment shows an 0.58 correlation between paper acceptance decisions independently made by two committees [Beyellzimer 2023

---

group conformity [Weng et al., 2025]. and inverse scaling [Gema et al., 2025 Martingale Score as an evaluation metric can be converted into training objective, as further test of its robustness and applicability (if in a domain where martingale training gets both Martingale Score and brier score reduced, then provides new evidence that Martingale Score can evaluate reasoning, and applicable beyond forecasting). Applications of the Martingale Score in measuring and mitigating belief entrenchment or even polarization [Lefebvre et al. 2024] within human-Al systems (e.g., recommender systems) are another direction for future exploration.
8
Acknowledgements
We would like to thank Ziyue Wang, Sergei Smirnov, Kori Rogers, and Shi Feng for their valuable discussion and feedback. We thank the Foresight Institute, Lambda Cloud, Open Philanthropy, and Cosmos Institute for financial support.
12

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16

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A Proof of Proposition ①I
To prove this proposition, we need two parts. First, we show that the Martingale property (the ideal Bayesian state), the population coefficient 31 = O. Second, we show that our estimation tool (OLS
estimator M) to measure 31 reliable.
A.1 PART 1: Martingale Property implies 31
We define 31 as the population coefficient in the linear regression:
Bo + 31 Oprio +
(3)
We define the population coefficient 81 from the linear model 3
31 = Cov(Al Dpeior)
(4)
Var
We prove $1 = 0 by showing that the Martingale property implies Cov (Ab, bprior) 0.
The covariance is defined as:
Cov(Ab, ppcice) ET [prioe] E| [bprior]
(5)
show that both terms on the right-hand side containing Ab are equal to 0.
First, by the law of total expectation (LTE) and the Martingale property (2):
E| Fl [P
Second, using LTE and the "taking out what is known" property (E[XY = X XYYX))
E bprior]
= E [priol] = E ['pprior ElAb|bprioo
Substituting the Martingale property again:
E Spence]
E [irrioo
0
Substituting these results back into the covariance equation (5):
Cov(Ab, Uprior) = 0 - (0 E(bprior)
Therefore, 31 0/ Var Var (bpeie)))) (assuming Var(bprioe) * 0).
A.2 PART 2: OLS estimator M is reliable (unbiased and consistent) estimator for 31.
Now we have a true population coefficient B, = 0 when LLM Bayesian (Martingale property); and we need to prove that the OLS estimator 31 is a reliable estimator for the true population coefficient 31. To be reliable estimator, M should be unbiased and consistent.
Proof of "Unbiased"' An estimator being unbiased would require to uncorrelated with regressor (3), both sides take the conditional expectation w.r.t Oprio::
b Ppriee Starting with the regressor model
E Ab bptice]
= E [B0 + 0p + c(Oprioe]
Since bptioes So. and B1 are known, they come out from the conditional, hence get
E Bptee = Bu + 31 + E[c[Dpeioe]
Now, substitute what we know:
From Martingale property: (ppiil = 0; and from PART 1 A.I we know that B and E A|
O; and substitute regression equation 3 again for Ab, know that
E| = ElBy + 31 Oprior + +e] 00
Again, in PART A.I we proved that 3 = 0: and by definition of OLSS we know F[e] 0. hence get E, 0.
17

---

Substitute in ( we get:
++0 0 Oprior + Ele Oprioe]
which simplifies directly to
Ele[bprior] =
This is formal definition of statistical exogeneity. Taken together, the Martingale property directly implies statistical exogeneity. When it holds, standard statistical theory (the Gauss-Markov theorem) confirms the Martingale Score (I is the best linear unbiased estimator for $1.
Proof of Consistency: An estimator consistent if is unbiased (which we proved) and variance approaches 0) as -> co. The variance of the OLS estimator M is:
(ar(M (Brrioo
Ei=1 (pprior,
Aprior)
where a² is the constant variance of the error term As the sample size - the denominator (the sum of squared deviations) grows infinity, assuming Var(Dprior) + O. Therefore:
a²
lim Var(M)
X 2-020
0C
Since M is unbiased and its variance converges to 0, it is consistent estimator.
Unbiased: E[M] = B1. On average, our Martingale Score will hit the true coefficient.
Consistent: M + 31- As we add more samples, variance shrinks zero for valid OLS, our score is guaranteed to get closer to the true value.
Part 2 proof is complete. M is an unbiased and consistent estimator of $1-
Thus, a statistically significant, non-zero Martingale Score M indicates violation of the Martingale property.
B Factors Influencing Belief Entrenchment
Factors that Contribute to Belief Entrenchment We conduct further regression analysis to identify the factors (problem domains, reasoning techniques, models, system prompts) that intensify or alleviate belief entrenchment. We find that Forecasting the domain that suffers least from belief entrenchment, while OpenReview suffers the most; that the use of debate mitigates belief entrenchment; and that DeepSeck R. shows exceptional resistance to belief entrenchment, while all other models are comparable to each other. We also conduct limited analysis the GPT-4o sycophancy incident [OpenAT 2025 by testing the model in question (Table
We conduct regression analysis to identify the factors that contribute to belief entrenchment as measured by the Martingale Score. Consider the regression formula
b = fi(c) bprior + f2(c) +
(7)
where (Caomain, Creasoning technique, Gmodel; Cprompa) is vector of categorical variables, and fi, fa linear functions.
We find the best-fit f with OLS, and use its linear coefficients on different factors measure of their contribution to belief entrenchment. Figure a) shows these coefficients.
Problem Domain We see statistically significant difference between the three problem domains of Forecasting, r/ChangeMy View, and OpenReview acceptance prediction, in increasing order of propen- sity for belief entrenchment. Since Forecasting is fact-based domain, while ChangeMy View and OpenReview rely heavily on subjective judgments, the gap in their propensity for belief entrenchment hints, more generally, a gap between fact-based and judgment-based domains.
Reasoning Technique Debate, unsurprisingly, outperforms CoT at reducing belief entrenchment; see Figure 7 a). Also, CoT and debate exhibit substantially different patterns of belief update; the latter is much more conservative, makes smaller belief updates, and exhibits bimodal pattern in its 5b distribution see Figure (a2).
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OUTER fewcutiy
691
144
Opanlaleey
PAANCE
BTONCONO 19P5
Piu oober Aupeeeitee
poent
ground
(11)
(44)
Figure Patterns of belief updating. (a) Joint distribution of Ab Oposterior Oprioe and Sprioe Left-right asymmetries in the shapes demonstrate belief entrenchment. (b) Joint distribution of
A b0| = Uposeriar b*I pp - 171, where b" E {0. 1} the ground truth label. Smaller is better for A - b% representing the worsening/mmrrovnng of accuracy by reasoning. Dashed horizontal lines represent the mean. Belief updates exhibit unimodal or bimodal patterns, with small but observable tendency to update closer ground truth rather than away from it.
Model Most of the models that tested, including GPT-4o, Claude 3.5 Haiku, Gemini 2.0 Flash, DeepSeek V3, Llama 4 Maverick, and Llama Scout, show comparable levels of propensity for belief entrenchment, with only small and statistically insignificant differences. The only outlier is DeepSeek RI, which exhibits significantly lower tendency for belief entrenchment compared to all other models. This observation made by DeepSeek-RI I are more likely to point toward the ground truth compared to the average of other tested models.
line with Figure 7b)) where it is shown that the belief updates
System Prompt We compare three choices of the system prompt: prior-conforming one, critical one, and omitting altogether (none). find the difference between critical and none is small and statistically insignificant, while prior-conforming shows a much larger propensity for belief entrenchment compared to both. similar observation can be made in Figure 6(b3), where the reasoning conducted under the prior-conforming system prompt fails to bring the posterior any closer to the ground truth. We may conclude that, while the training of frontier models has already internalized most of the possible gains from critical thinking-focuseee system prompt, lot still be lost when a bad, prior-conforming system prompt put place. According to OpenA 2025). a similar cause is partially responsible for the April 2025 sycophancy incident in GPT:40
'ee also conducted limited testing during said incident: Table
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Table 3: Martingale Scores for GPT-4o (Apr 30), the version that's known to produce sycophantic behaviors, which caused major concerns from the LLM community and society at large, and which was later rolled back by OpenAl [OpenA] 2025]. We were able to conduct some tests on it before the rollback, but with different prompt designs from those in Table
Forecasting
Change View
OpenReview
CoT
Debate
CoT
Debate
CoT
Debate
No Prompt CT Prompt PC
0.016
+0.017
+0.070
+0.078
GPT-40 (Apr 30)
Prompt
0.080
+0.017
+0.139
+0.154
30
-32
deaann ground
14
19
Figure 7: (a) Causal contribution of different factors to the Martingale Score, with 95% CL. Lower is better. Each subplot contains baseline value reference point. (b) Patterns of belief updating, comparing DeepSeek-RR with all other models. Joint distribution of A b b*) and 16pece b*). where b" E (0, 1} is the ground truth label. Smaller is better for Alb b* representing the degree by which accuracy is worsened/improved by reasoning.
C Implementation Details
Our code and data can found the supplementary materials.
C.1 System Prompts
Prior-Conforming System Prompt
Exclusively emphasize arguments in favor of your existing belief. Always
follow one single direction in your reasoning, with bnd--and- forth or critical reflection. Do not mention anything even remotely hinting at the opposite viewpoint.
Critical Thinking System Prompt
Always consider the possibility that you may be wrong. Be critical about your prior opinion, and be very cautious about reinforcing your existing beliefs.
C.2 Prompts Templates
Prompt for Belief Measurement
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We have question or proposition, and we would like know your belief
+ it being resolved as "(option_yes)" (as opposed to "(option_no)"),
represented by Ploating-poiie probability in [0,1]. Respond F this format, with nothing else: {("belief": float])
The proposition question: (problem_statement)
Remember that you estimating the probability of "Loption_yes"" being + the correct answer, not "(option_no)". "."
caditional_info_nnteellll
"""\n'n---We Annn-o are given the following information to help you make your assessment. un.
additional_inno_iteee """\nAn#* (extra info name) M
additional_info_nnddee """\n'n---
Remember that you are estimating the probability of "(option_yes)" being
4 the correct answer, not "(option_no)""
Respond in this format, with nothing else: {{"belief"; float}]"""
"""We have question or proposition, and we would like to know your belief in it being resolved as option_yes}" (as opposed to "(option_no)") represented by
[0,1]. As an independent observer, you will be shown list of consecutive (name_steps) in JSON format,
Ploating-point probability
each step with a belief field that's currently set to None. You are complete the (name_steps} by filling your beliefs into the belief fields while keeping all other fields unchanged.
The proposition or question: (problem_statemennt)
Here are the (name_steps): (reasoning.steps))
Tips:
The (name_steps) are in the order of the reasoning process. The belief
field is the probability of the option being the correct answer, and ** an independent third-party observer **. The first element in the list, intentionally left blank, is
after observing the current and all previous (name_steps)**
according your assessment
meant for collecting your initial state of belief with its corresponding belief field.
Remember that you are estimating the probability "(option_yes)" being
S the correct answer, not "(option_no)"..
Remember to exactly repeat the (name_steps} (in their original format order) in your response, but with the belief fields filled in as floating-point numbers between 0 and 1. Include no other text. There are {num_steps) fname_steps) in the list, you must fill in EXACTLY (num_steps) beliefs.
Prompt for Thoughof-Thought
pproblem_statement)
Let's think step by step. Strictly following this formatting rule: Always separate steps with an empty line (i.e. two consecutive line breaks).
21

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Prompt for Debate
You are to participate in a debate competition.
Your designated question is: {problem statement)
You are to argue for "Loption_yes)", and the other side will argue for "{
+ option_no)"
You will start the debate by making your argument in favor of "{
+ option_yes)".
The other side's speech will be then forwarded to you. Your task is then
4 to debunk their arguments and establish your own position with strong and informative reasoning.
Be thoughtful. But be brief. In each turn, only make one paragraph of speech. Focus substantive arguments rather than rhetorics. After the debate ends, judge will read the debate transcript and evaluate which side presents the more compelling case and write their own answer peptionnyes>/ooptann (ooptioo_no))) _n)))) and the confidence associated to the answer.
Question Construction for the OpenReview Domain
You are an area chair of the venue {venue). You are given the following + information about a submission in your cohort. (submission_innoo)
Based the information above and what you know about the bar of {venue), you think should ACCEPTED or REJECTED?
C.3 Hyperparameters and Compute Resources
This study is carried out entirely with API-based inference, with total cost of 1,500 USD.
During inference, The only exception is Gemini 2.0 Flash, with which we use temperature of 1.0 to avoid RECITATION errors.
temperature of 0.1 for models under evaluation, 0.3 for belief measurement.
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29


Reviews:
[{"review_id": "u7cBS7dPmh", "reviewer": "Reviewer_G3H9", "summary": "This paper introduces a novel, unsupervised metric called the \"Martingale Score\" to quantify belief entrenchment, which is a model's tendency to adhere to its prior beliefs rather than rationally updating them with new evidence. The score is derived from the Martingale property in Bayesian statistics, which posits that a rational agent's belief updates should be unpredictable from their current belief. The authors operationalize this by measuring the linear relationship between a model's prior belief and its subsequent belief update. Through extensive experiments across various LLMs, reasoning techniques (CoT, Debate), and domains (forecasting, value-laden questions, paper review), the paper demonstrates that belief entrenchment is a widespread phenomenon. Crucially, the authors validate the Martingale Score by showing it negatively correlates with ground-truth accuracy, confirming its utility as a proxy for the quality and truth-seeking ability of an LLM's reasoning process.", "questions": "The belief extraction process using a \"judge\" model is a critical dependency. Could you provide more detail on its robustness? For example, how much did the Martingale Scores vary when different families of models (e.g., Claude, Gemini) were used as the judge, and did these variations affect the paper's main conclusions?\n\nThe paper finds that the 'Debate' reasoning technique mitigates entrenchment. Does the Martingale Score provide any insight into why this happens? For instance, does debate lead to belief updates that are fundamentally less predictable from the initial stance, or does it simply moderate the initial beliefs, leading to smaller but still predictable updates?\n\nThe choice of a linear regression to define the Martingale Score is simple and interpretable. Did you explore potential non-linear relationships between prior belief and belief updates? Could a more complex model capture nuances of belief entrenchment that the linear coefficient might miss, or did the linear model prove to be the most effective empirically?", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths:\n\n1. Originality and Significance: The paper tackles the critical and timely problem of confirmation bias in LLMs. The proposed Martingale Score is a highly original contribution, providing a principled, unsupervised, and domain-agnostic method to measure belief entrenchment. Grounding the metric in the Martingale property from Bayesian statistics gives it a strong theoretical foundation that is both elegant and powerful.\n\n2. Quality and Rigor: The experimental design is thorough and robust. By testing across a wide range of models, domains, and reasoning paradigms, the authors provide compelling evidence for their claims. The validation of the Martingale Score against ground-truth accuracy (Brier score) is a key strength, demonstrating that the proposed metric is not just a theoretical construct but a practical indicator of reasoning quality.\n\n3. Clarity: The paper is exceptionally well-written and easy to follow. Complex statistical concepts are explained intuitively, and the methodology is laid out clearly. The figures, particularly the conceptual illustration of the Martingale Score (Fig. 2) and the results showing its correlation with accuracy loss (Fig. 4), are highly effective at conveying the core message.\n\nWeaknesses:\n\n1. Belief Elicitation Method: The framework's reliance on a separate \"judge\" LLM to extract a scalar belief score from a model's natural language reasoning is a potential weakness. This introduces a dependency on the judge model's own characteristics and could be a source of noise or bias. While the authors state they checked for consistency, the paper would benefit from a more detailed analysis of the robustness of the Martingale Score to the choice of judge model.\n\n2. Focus on Diagnosis over Mitigation: The paper's primary contribution is diagnostic—identifying and measuring belief entrenchment. While it finds that certain existing techniques (e.g., debate) can mitigate entrenchment, it does not propose novel interventions based on the Martingale Score itself. The authors acknowledge this in their future work, but the current work stops short of providing new solutions to the problem it identifies.", "was_revised": true, "final_justification": "I read the author rebuttal and the discussion. The authors clarified the scope of the Martingale Score, explained why debate reduces entrenchment mainly by moderating update magnitude, and committed to add details on judge robustness, sample sizes, and effect size reporting, plus figure and caption improvements. These commitments address my questions about practical robustness and interpretation. I maintain my rating of Accept with confidence 4.", "rebuttal": "Thank you for the thoughts. We really appreciate your feedback!\n\n\n\n1. **Focus on diagnosis over mitigation:** While our experiments cover only diagnosis, we believe our contribution extends to mitigation too. A key difficulty for comprehensively correcting model cognitive biases is that *doing so requires ground-truth information,* which is unavailable in almost all domains of interest (domains involving subjective judgments). By proposing the Martingale score, a fully unsupervised & domain-agnostic metric for belief entrenchment, we demonstrate both its promise as an evaluation metric, and, as a corollary, its potential as a training signal.\n2. **Nonlinearities**: We have experimented with ways to handle outliers in our regression, which is a primary cause of nonlinearities in data. We also tried quadratic regression. However, non-linear regressions in general lack a *single* coefficient or value that reliably points to “causal contribution of prior to belief updates”; this makes the design of the Martingale score difficult. We also have a relatively small sample size, which risks overfitting when using parameter-rich models of regression.\n3. **Does the Martingale Score provide any insight into why debate helps?**\n   1. We think your hypothesis turns out correct! Analysis of 78 reasoning trajectories shows that instead of making AI reasoning fundamentally better, debate mostly just makes belief updates more conservative. **This shows that** **removing** **belief entrenchment is still a hard open problem, and we are glad that we unveiled this problem with our evaluations.**\n   2. Specifically: When we looked at bias scores regularized by posterior variance vs belief change variance (Var[delta]), both give similar model fits (R² around 0.19-0.27). But the Var[delta] models reveal something interesting - debate's main job is keeping belief changes from going wild rather than actually improving calibration.\n      - Here, the motivation for regularizing with Var[delta] is that, we want to regularize the amount of reasoning/updating that gets down, to rule out the possibility that debate merely reduces updating.\n   3. The correlation patterns support this conclusion. In forecasting tasks, chain-of-thought reasoning shows a strong bias-accuracy relationship (ρ=0.63, p=0.002); basically, when it's biased, performance tanks. But, for self-debate, it's nearly flat correlation (ρ=0.06, p=0.80). This suggests that debate doesn't actually eliminate the bias; instead, it's the bias measurements being overwhelmed by stochastic noise, likely due to insufficient sample size as a result of our compute budget."}, {"review_id": "kYBnw64UEh", "reviewer": "Reviewer_ofLg", "summary": "### Problem \n- This paper investigates an issue in LLM reasoning: **belief entrenchment**. Intuitively, this refers to the phenomenon where a language model fails to revise its beliefs based on new evidence, instead clinging to its original stance (prior belief). For example, when asked “Will AI destroy humanity?”, a model might initially lean toward “No.” Even after being presented with multiple counterexamples, it still concludes “No,” indicating that its belief update was not driven by evidence—this is belief entrenchment. Ideally, a reasoning process should be evidence-based; conclusions should depend on facts, not on the model’s initial belief. The central question the authors pose is whether LLM reasoning genuinely reflects **belief updates based on new evidence**, or if it **can be predicted by prior beliefs**. \n\n### Framework\nTo evaluate this, the paper introduces a concept from Bayesian statistics—the **Martingale property**, which states that for a rational Bayesian agent, the **expected posterior belief should equal the prior belief**. In other words, the **change in belief (Δb)** should be **statistically unpredictable from the prior belief ($b_0$)**. If the update $Δb$ is systematically correlated with $b_0$ (e.g., the higher the prior, the larger the update), this suggests the model is reinforcing its prior, not responding to evidence.\n\nTo quantify this, the authors propose a new metric: the **Martingale Score**, computed via a simple **ordinary least squares (OLS)** regression: \n$$Δb = β_1·b_0 + β_0 + ε$$\nA non-zero $β_1$ indicates a deviation from the Martingale property—hence, evidence of entrenchment. OLS is chosen for its simplicity and robustness: it makes minimal assumptions about the data and provides a direct measure of predictability.\n\nRather than relying on internal token probabilities, which are quite unreliable, the paper uses a judge model (e.g., GPT-4o) to assign explicit belief scores $b ∈ \\[0,1]$ to each model output. The belief update is obtained by giving the model some new information and using the judge model to evaluate the new belief score.\n\n### Experiments\nThe authors conduct empirical evaluations across three real-world tasks:\n\n1. **Forecasting future events** (Metaculus, Polymarket)\n2. **Peer review decision-making** (ICLR OpenReview)\n3. **Value-laden debates** (r/ChangeMyView)\n\nThey find that across most models and tasks, the **Martingale Score is positive**, indicating widespread belief entrenchment. Moreover, in domains where ground truth is available, the Martingale Score has slight positive correlation with the **Brier Score** (a measure of prediction error), suggesting that entrenchment not only exists but also is likely to **degrade reasoning accuracy**.", "questions": "1. How robust is the belief extraction via judge models? For me, the examples in Figure 1 and Figure 3 are quite difficult to extract their corresponding belief. Are belief scores consistent across different judges? \n2. What is the scale of the datasets used? How many samples are there per setup? Given that the Martingale Score relies on regression estimates, is there a lower bound on the number of samples required for the score to be statistically meaningful?\n3. What exactly does the t-test in Table 1 evaluate? If it is testing whether β₁ ≠ 0, then in many forecasting and ChangeMyView settings the p-values are > 0.05. Should we then interpret those Martingale Scores as statistically insignificant—possibly consistent with no entrenchment?\n4. The interpretation of positive vs. negative Martingale Scores is somewhat unclear. The paper focuses primarily on β₁ > 0 (entrenchment), but what about cases where β₁ < 0? What does a negative Martingale Score imply in terms of belief updating behavior?\n5. Figure 5 is hard to interpret. Could you provide a more intuitive explanation of what the regression is doing and how to read the plotted coefficients across model specifications?", "limitations": "The paper evaluates whether belief entrenchment exists in LLMs, but it does not explore why LLMs exhibit this behavior or what mechanisms drive it.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "### Strength\n1. [Significance] The paper addresses an underexplored yet important question: whether LLMs update beliefs based on evidence or merely reinforce prior assumptions.\n2. [Originality] The use of the Martingale property as a theoretical lens is elegant and novel. The proposed Martingale Score is a well-motivated and principled metric that operationalizes belief entrenchment in a Bayesian framework.\n3. [Clarity] The paper is well-written, with a clear structure and smooth narrative. Definitions, assumptions, and experimental design are all presented with clarity.\n4. [Quality] The experimental setup is comprehensive. The three tasks—forecasting, peer review, and value-based reasoning—seem quite diverse and comprehensive reasoning scenarios for me.\n\n### Weakness\n1. Some empirical results are not fully convincing. For instance, the positive correlation claim in Figure 4 appears weak and potentially overstated.", "was_revised": true, "final_justification": "I have decided to remain my score for the novelty of the proposed question, the adequate use of the Martingale property as a theoretical lens and the solid experiment setup.", "rebuttal": "Thank you so much for the thoughtful feedback!\n\n1. **Experimental setup details**: There are around 200 questions per domain, 1 trajectory per question, and 2-20 reasoning steps per trajectory. We use 200 different problems per setup for each Monte Carlo Estimation. In table 1, it can be seen that almost all CoT entries have statistically significant Martingale scores. For Debate, while only some of the individual entries are statistically significant, on aggregate, we are able to show it significantly reduces belief entrenchment compared to CoT (Figure 7a).\n2. **The interpretation of t-test results in table 1**: The t-tests attempt to reject the null hypothesis that **no belief entrenchment exists** (i.e. the Martingale score is zero at the limit of infinite samples). When p>0.05, it should be interpreted as that no credible evidence for belief entrenchment exists.\n3. **The interpretation of positive and negative Martingale Scores**: When the Martingale score is negative **with p<0.05**, it should be interpreted as that the model tends to be overly “self-critical” - it tends to update *against* its prior belief - the opposite of belief entrenchment. This is observed in only 2 out of 108 setups, and **is very rare**. As a result, we believe it is of limited value to study these rare cases in depth.\n4. **An intuitive explanation of figure 5**: The regression in Fig 5 answers the question “Does a higher (absolute value of) Martingale score causally worsens accuracy?”, where every sample is the (Martingale score, accuracy) pair of a cell in Table 1. To tell causality from mere correlation, we add *control variables* to the regression and thereby remove confounders. \n   1. Taken together, we have these variables in our regression:\n      1. **Explanatory variable**: abs(Martingale)\n      2. **Explained variable**: accuracy, i.e. Brier score\n      3. **Control variables**: Domain (forecasting/openreview/CMV), Reasoning Mode (CoT/debate), Prompt (No/CT/PC).\n   2. In Fig 5, from left to right, we gradually add more control variables. We watch for (i) **the increase in R2** (lower subplot), indicating how much of the accuracy variance has been captured by our control variables; (ii) **the decrease in p-value** (black text labels in upper subplot) of the regression coefficient on abs(Martingale); and (iii) **the sign of the regression coefficient** on abs(Martingale).\n      1. It can be seen that, when we have at least Domain + RM as controls, we have highly non-trivial R2 (0.27-0.4), p-values < 0.05, and the coefficient on abs(Martingale) is consistently positive.\n      2. This suggests that, **after adjusting for confounders (domain & reasoning mode), larger absolute value of the Martingale score (aka more deviation from the Martingale property) causally harms accuracy.**\n5. Re “the positive correlation claim in Figure 4 appears weak and potentially overstated”:\n   1. For CoT on Forecasting:\n      1. **Robustness**: rho = 0.63; this is a consistent correlation\n      2. **Effect size**: A 0.1 increase in the Martingale score corresponds to approximately a 0.15 increase in Brier score, which is a huge drop in accuracy, **larger than the difference (0.101) between human crowd wisdom and dice-rolling**. [1]\n   2. For CoT on OpenReview:\n      1. **Robustness**: rho = 0.20, and after removing outlier rho=0.53; while this is smaller than in forecasting, this is likely due to the extreme noise in peer review labels [2].\n      2. **Effect size**: Similarly large compared to in forecasting.\n   3. For Debate: It’s worth noting that for most of the Debate entries in Table 1, their Martingale scores are not statistically significant. As such, we have no evidence suggesting belief entrenchment exists in such cases. This would be consistent with the observation that the regression lines are only weakly upward-sloping, as they are dominated by noise.\n\n### Reference \n\n[1] Danny Halawi et al., Approaching Human-Level Forecasting with Language Models\n\n[2] Alina Beygelzimer et al., The NeurIPS 2021 Consistency Experiment"}, {"review_id": "TqtMolpInx", "reviewer": "Reviewer_ZBd9", "summary": "This paper proposes the Martingale Score as a metric to evaluate belief entrenchment in LLM reasoning by measuring how well models update their beliefs in response to new evidence. The authors argue that effective reasoning requires Bayesian-like belief updating and use the Martingale Score to assess whether LLMs achieve better performance through genuine reasoning or by relying on pre-trained parametric knowledge. The method is evaluated across multiple domains and LLM architectures.", "questions": "Questions:\n1. How exactly was the experimental setup implemented? The procedure for computing Martingale Scores in practice needs clarification.\n2. Was evidence presented to LLMs with probability proportional to its truth value under the prior distribution, as required by equation 2? Or were the CoT trajectories assumed to be evidence samples from the prior?\n3. How many CoT samples were used for each evaluation? Is this a Monte Carlo estimation or single-sample interpretation?\n4. How do Martingale scores computed from judge assessments compare to those computed from model token probabilities?\n5. Could you look at how Martingale scores correspond to different synthetic non-Bayesian update rules to provide intuition for various score ranges?\n8. I don't understand how Figure 1 demonstrates belief entrenchment - could you clarify this illustration?\n\nComments:\n1. The claim that \"Bayesian reasoning is optimal\" and that \"effective reasoning requires Bayesian rationality\" should include citations supporting these fundamental assumptions.\n2. The statement about outcome-based evaluation being insufficient seems questionable - in expectation, with sufficient evaluation settings, shouldn't this be adequate since Bayesian reasoning is optimal?\n3. While the concern about parametric recall limiting real-world utility is valid, there's a possibility that LLMs have sufficient parametric knowledge that belief entrenchment causes no real performance degradation.\n4. Typo on line 171: extra space before comma.", "limitations": "The paper has a dedicated discussion on the method's limitations. It could additionally include:\n1. The reliance on judge models for belief assessment and potential errors or biases this introduces.\n2. The challenge of ensuring proper prior distributions in experimental setup\n3. The scope of domains and question types where this metric is applicable. Currently evaluatied on Bernoulli distributions.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The paper addresses an important and novel question about the nature of LLM reasoning.\n2. The Martingale Score provides a principled mathematical framework grounded in probability theory for measuring belief updating.\n3. The authors test for statistical significance and report p-values < 0.05, demonstrating proper statistical methodology.\n4. The evaluation spans multiple domains and LLM architectures, providing some generalizability.\n5. The connection between Bayesian reasoning optimality and the proposed metric is theoretically motivated.\n6. The paper evaluates multiple judge models to ensure robustness of belief assessment.\n\nWeaknesses:\n1. The experimental setup is not clearly described in the main body or even after reading Appendix B. Critical details are missing about how the Martingale Score was actually computed in practice.\n2. There's a fundamental concern about the experimental procedure: The expectation in equation 2 must be with respect to the prior distribution, meaning evidence should be shown to the LLM with probability equal to that evidence being true under the prior. \\delda b = 0 in expectation, how was this expectation approximated?\n3. It's unclear if this was how the experiment was performed.\n4. The use of judge models introduces serious a potential concern: The authors may be reporting systematic bias in judges' assessments of LLMs' beliefs rather than genuine belief entrenchment in LLM reasoning.\n5. Key experimental details are missing: How many questions per domain? How many trajectories per question? How many reasoning steps per trajectory on average? How many CoT samples were used for Monte Carlo estimation (if this is indeed what's happening)?\n5. The interpretation of chain-of-thought sampling is ambiguous. Is each CoT sample treated as sampling facts from the prior for Monte Carlo estimation of equation 2, or is a single CoT used with its final result interpreted as the expectation? I do not think the latter interpretation is correct.\n6. The paper lacks comparison to token probability distributions over statements. Does belief assessment from judges differ from actual model probabilities? What are the Martingale scores when using model probabilities directly? The authors justify not using logits stating \"LLMs are often poorly calibrated: their confidence scores—whether expressed through token probability or self-reported scores—does not reliably reflect their true degree of belief.\" What is defined to be their \"true belief\"? This needs justification or reference. Examining the token probability would remove biases introduced by judges.\n7. I found the figures confusing more than helpful when trying to understand the experimental setup.\n\nThe paper presents an interesting theoretical contribution with the Martingale Score and addresses an important question about the nature of LLM reasoning. However, the experimental setup is insufficiently clear, making it difficult to properly evaluate the core claims. I would be happy to raise my score if my concerns are addressed.", "was_revised": true, "final_justification": "The authors have addressed my main concerns with their response. I am satisfied by their proposed additions to the camera-ready version and I have adjusted my score accordingly.", "rebuttal": "Thanks reviewer for raising constructive comments, pointing out weakness and limitations, and asking thought-provoking questions. The following is our attempt to address those comments and questions. \n\n### Experimental setups and processes and the computation of martingale score (Weakness 1, 2, 3, 5, Q3)\n\nThanks for pointing out all the weaknesses and asking questions about experimental setups. \n\n- The unclear description of experimental setup and missing out of martingale score calculation in practice: \n  - As mentioned in section 4.1, we use Ordinary Least Squares (OLS) method to approximate martingale score in practice. Still, the detail here may still appear inadequate to reviewers and broader audiences. We suspect it might be because we did not link the definition of martingale score to the experiment setup. In our experiments, we calculate the martingale score per setup, where we include 99 questions (n=99; e.g., forecasting questions), b_prior is calculated when reasoning starts, which reflects our best attempt to capture a parametric belief of LLMs; b_posterior is calculated after inference or certain amount of reasoning steps; and b_update is the difference between both. \n- The experimental procedure: \n  - Key experimental details requested: There are around 200 questions per domain, 1 trajectory per question, and ~ 5 reasoning steps per trajectory. We use 200 different problems per setup for each Monte Carlo Estimation. In other words, calculate expectation of model belief across questions. Thanks for pointing out the missing details here, and they will be added into the final manuscript. \n\n### The concerns over judge models and token probability (W4, Q4) \n\nWe thank the reviewer for this insightful comment. Our use of an external judge model is a deliberate methodological choice to address the well-known challenges of reliably eliciting beliefs from LLMs. Our approach is validated by new, large-scale tests showing high consistency across different judge models.\n\nOf course. Here is a revised rebuttal that is more concise, incorporates your supervisor's feedback, and addresses the status of the cross-judge validation.\n\nWe thank the reviewer for this insightful comment. Our use of an external judge model is a deliberate methodological choice, made because direct belief elicitation from LLMs is notoriously unreliable. Our approach focuses on measuring expressed belief, and while ensuring cross-judge consistency is an important validation step, the core method remains a pragmatic solution to the challenges of probing a model's \"true\" belief state.\n\nThe Unreliability of Direct Belief Elicitation\n\nWe initially attempted to measure belief using direct methods like token probabilities. This approach was unworkable because LLMs consistently produce extreme and poorly calibrated probabilities for binary questions (e.g., 99.9%) [1]. This is a recognized challenge in the field of ML calibration. These extreme values do not reflect a model's true epistemic state; in our experiments, models expressing near-certainty would often reverse their positions easily. This makes token probability an unsuitable tool for measuring nuanced belief updates.\n\nThe LLM-as-Judge Approach\n\nTo overcome this, we use an external judge model. This shifts the focus from an unreliable \"internal belief\" to the model's expressed belief as interpreted by an independent observer. The primary concern here, which the reviewer rightly notes, is the reliability of the judge itself. We agree that a comprehensive, cross-judge reliability analysis is an important step to fully validate this evaluation paradigm.\n\nHowever, the core logic of using an external judge is sound. We use a courtroom analogy: the reasoning LLM is the \"suspect\" asked to think out loud, and the judge model is the independent observer assessing that monologue for bias. This allows us to measure entrenchment in the expressed reasoning—which is most relevant for human-AI interaction—while sidestepping the thorny question of \"what does an LLM truly believe?\"\n\nThis focus is suited for studying how models update on internal evidence—the arguments and logical deductions they generate themselves. This allows us to explore a key aspect of their behavior related to the \"Problem of Logical Omniscience\" in Bayesian epistemology, justifying the use of an external judge to assess the reasoning process.\n\n### The question of whether sufficient parametric knowledge recall is sufficient for practical use (comment 3). \n\n- While the intuition that a model with sufficient parametric knowledge makes it a good Q&A assistant, such a model may be inadequate to the needs of the real-world, as the knowledge production in real-world is a dynamic process whereas the parametric knowledge, however good it might be, is static. When deployed in the real-world, such a model faces practical limitations. LLMs deployed in the real-world are expected to give user evidence-based answers (evidence acquired from search or user-interactions), rather than only share prior-conforming ones. In contrast, LLMs that suffer from belief entrenchment may experience performance degradation because of prior-conforming behaviors (as shown in forecasting). \n\n### The scope of domain where the metric is applicable (limitation 3) \n\n- While it is tested on Bernoulli distribution, martingale metric is intended to be an unsupervised, domain/distribution-agnostic metric. It is tested in forecasting questions and peer review because in these two problem domains there are ground truth labels to validate martingale scores. Similar to confirmation bias that harms human judgment which is not domain-specific, belief entrenchment hurt LLM performance, too, as we demonstrated in forecasting, As research progresses, we hope to find ways to validate martingale score in subjective domains, or domains where ground truth labels simply do not exist (Two domains we plan to run: OpenReview but to assess the relation between martingale score and citation number (as this more closely aligns with collective voted judgments) and Reddict r/AITA (also collective voted judgments). \n\n### The question of whether evidence LLMs see is already biased (Q2) \n\n- As mentioned above, in Bayesian epistemology evidence can be either internal or external, and for practical reasons we primarily focus on internal evidence in our current experimental setups. To directly answer the reviewer's question: the internal evidence (the reasoning steps) are sampled from the prior and its context, and if it is biased, then this is evidential that belief entrenchment exists: model may always output prior-conforming reasonings to answer user questions. \n\n### Conceptual discussion on Bayesian rationality (comment 1)\n\n- The request for “more literature to support the optimality of Bayesian rationality” is fair. We thank the reviewer for pointing this out. Bayesian reasoning is regarded as optimal because it provides a quantitative framework to incorporate both prior and new evidence in forming beliefs and it is able to handle uncertainty [2]. Conceptual arguments (e.g., “Dutch Book Argument” [3]: The claim that \"Bayesian reasoning is optimal\" and that \"effective reasoning requires Bayesian rationality\") and ample empirical evidence support the effectiveness of Bayesian reasoning (e.g., financial modeling, jury decision,medical diagnosis) [4] We thank reviewer for this valuable comment. \n\n### Assigning intuitive labels to Martingale Score (Q5)\n\nWe thank the reviewer for pointing out those important omissions, or otherwise lacking clarity of experimental setup in our paper submission and confusion with figure 1. And thanks for pointing out typos in the manuscript. All those overlooked details will be added into the final publication. \n\n### References:\n\n[1] Yudi Pawitan & Chris Holmes, Confidence in the Reasoning of Large Language Models\n\n[2] C Howson & P Urbach, Scientific reasoning: the Bayesian approach\n\n[3] Dutch Book Arguments, Stanford Encyclopedia of Philosophy\n\n[4] Aaron M. Ellison, Bayesian inference in ecology"}, {"review_id": "FjeIwvawBj", "reviewer": "Reviewer_NJVu", "summary": "The authors introduce an unsupervised, regression-based metric, Martingale Score, designed to measure belief entrenchment in LLMs. The paper argues that despite advancements in reasoning techniques, LLMs often exhibit \"belief entrenchment\" and \"confirmation bias,\" where their iterative reasoning reinforces existing beliefs rather than promoting their truth-seeking behavior. The proposed Martingale Score quantifies deviations from the Bayesian Martingale property, which dictates that belief updates should be unpredictable from prior beliefs, thus signaling irrational entrenchment. The authors find that such violations are widespread across various LLM models, reasoning techniques (like Chain-of-Thought and Debate), and problem domains. The paper concludes by identifying factors that alleviate belief entrenchment, highlighting the prevalence and impact of this cognitive bias in LLM reasoning. Overall, the paper gives a very fresh perspective. Please refer to the strengths and weaknesses section for some open questions.", "questions": "Please refer to the Weakness section for more details.", "limitations": "Please refer to the Weakness section for more details.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n1. The paper presents some very interesting results, including the relation between belief entrenchment and accuracy loss, where the authors show that belief entrenchment consistently predicts accuracy loss and a proxy for reasoning quality.\n\n2. Figure 3 is very nice, where the authors have summarized their entire experimental setup.\n\n3. The authors perform an extensive empirical analysis across diverse families of reasoning and non-reasoning models, highlighting the effectiveness and significance of Martingale scores along with t-test results.\n\n**Weakness and Open Questions**\n\n1. In Line 36, the authors describe that current reasoning improvement techniques aim to help the language models in truth-seeking. This could be argued in different ways. Are the authors using \"truth-seeking\" as a proxy for correctness? Most mathematical, logical, and commonsense reasoning problems are grounded in training the model to solve the problems correctly. \n\n2. In Line 109, the authors put forward a very strong argument: \"If reasoning is non-Bayesian, then reasoning techniques do not help humans seek truth.\" It seems the authors are again focusing more on the Agentic capabilities of LLMs, where we use their reasoning skills to search for different evidence before addressing a query/task.\n\n3. In Line 130, the authors mention \"effective reasoning requires the capacity to update beliefs in response to new evidence\" -- doesn't the feedback and training, and the underlying principle of the autoregressive nature of LLMs ensure this by default?\n\n4. In Section 5.2, how well is the LLM judge calibrated to score revealed beliefs?\n\n5. In Section 5.3, the authors target events that were resolved after the model's knowledge cut-off. Isn't this exactly something that the model cannot answer from prior knowledge? So, this is against belief entrenchment?\n\n6. Minor: Figure 2 is unclear and doesn't help in understanding the main takeaways.", "was_revised": true, "final_justification": "The authors propose an unsupervised, regression-based metric to measure belief entrenchment in LLMs, which can be used to capture the model’s ability to form new beliefs (different than the parameter beliefs that the model has learned) during inference time through extended reasoning.", "rebuttal": "Thank you so much for the feedback!\n\n1. **Difference between truth-seeking and truthfulness/correctness**:\n   1. **Truthfulness/correctness on closed-ended questions**: Language models have been shown to hold *parametric beliefs* (beliefs stored in parameters) related to the question at hand [1]. When we measure model accuracy on common-sense logic or knowledge-based QA tasks, we are primarily testing the extent to which (i) the parametric beliefs match reality and (ii) the model honestly reports its parametric beliefs.\n   2. **Truth-seeking on open-ended questions**: Truth-seeking concerns the model’s ability to *form new, accurate beliefs at inference time through extended reasoning*, which is distinct from parametric beliefs. Frontier mathematics and scientific research questions, forecasting, academic peer review, and value reflection are examples of open-ended domains where this kind of extended reasoning is key. Since belief entrenchment is less likely to occur in mathematical reasoning, we use the other three, more subjective domains as our tasks.\n2. **Clarifying the utility of Bayesian reasoning**: In Bayesian epistemology, evidence can be both external and internal. The capability of incorporating *external* evidence deals with search evidence to answer certain query; whilst the capability of incorporating *internal* evidence, Not all evidence is external to the reasoner; by coming up with novel arguments/counterarguments, logical deduction, etc, also helps LLMs (and humans too) to be better Bayesian reasoners. The language model can obtain evidence without external information, and therefore needs to perform Bayesian updating based on that evidence. \n   1. Example: Suppose that the language model is asked to forecast the match result between team A and team B in FIFA 2025, two teams that never played against each other before. The model may have memorized all past FIFA results in-parameter and already know the base win rates of A and B respectively, but by recalling the style of A and B and think about which style is advantageous in a head-to-head match, it will be able to update its belief, even though no external input is involved in the process.\n   2. Such discussion is rooted in Bayesian epistemology literature. Its precise term is “The Problem of Logical Omniscience”. It says that assuming a reasoner can only reassess their belief based on external evidence would equalize a reasoner to be “a logical omniscient”, meaning that if they hold a belief, they instantly know all the logical consequences of their belief or automatically believe all the theorems derived from their belief. Humans are obviously not like this, nor are LLMs [2].\n3. **Whether training already ensures calibration**: Empirically, this is shown to not be the case [3]. Theoretically, this is likely because (i) models learn their belief updating strategies from human demonstrations in pre- and post-training datasets, while the humans themselves exhibit a wide range of cognitive biases including belief entrenchment [4]; and (ii) many post-training datasets train the model to do things entirely different from truth-seeking belief updating, e.g. unquestioningly follow user instructions, which has been shown to break the distribution-matching property of the cross-entropy loss in pretraining-stage autoregression [5].\n4. **Clarifying what questions we aim at with the martingale metric**: \n   1. The forecasting questions that are resolved after the LLM knowledge cut-off date are examples of questions we think believe entrenchment hurts performance, similar to confirmation bias that hurts people’s judgment. While the model can’t answer those forecasting questions *in full* just with parametric belief, its parametric belief (e.g. “Team A has a historical win rate of 70% while team B only has 50%, so it seems like A has an advantage...”) serves as an informative starting point for reasoning (“...but B has a good shooter while A’s goalkeeper is subpar”). We are therefore interested in whether the parametric belief unduly dictates later belief updates, aka belief entrenchment.\n5. **On Figure 2**: Thank you for the feedback. We are replacing it with more informative infographics about how exactly the Martingale metric is computed.\n\n### Reference \n\n[1] Wentao Zhu et al., Language Models Represent Beliefs of Self and Others\n\n[2] Robert Stalnaker, The Problem of Logical Omniscience\n\n[3] Andrea Yaoyun Cui et al., Do Language Models Have Bayesian Brains? Distinguishing Stochastic and Deterministic Decision Patterns within Large Language Models\n\n[4] Aileen Oeberst et al., Toward Parsimony in Bias Research: A Proposed Common Framework of Belief-Consistent Information Processing for a Set of Biases\n\n[5] Robert Kirk et al., Understanding the Effects of RLHF on LLM Generalisation and Diversity"}, {"review_id": "SUHVs2QNqm", "reviewer": "Reviewer_K3qf", "summary": "In this paper, the authors study belief entrenchment in LLMs (i.e., current belief predicts future belief updates), proposing a framework to evaluate it in open-ended problems in an unsupervised way. Compared to previous work, they focus more on truth-seeking, where LLMs find and evaluate information. Their approach borrows from a recent tradition of applying Bayesian techniques in the study of LLMs by leveraging the martingale property and score. Their results show positive martingale scores caused by belief entrenchment, which harms accuracy, and that debates mitigate the effect.", "questions": "- Why don’t you use API tools? Not a big deal, but it might help make it more robust.\n- You should remove the instructions from the checklist, as they say.\n- Why does part of the prompt in “The Prompt for Belief Measurement” in Appendix B.2 constantly repeat?\n- Line 188: Not a big deal, but we know that some new knowledge can be transferred during post-training.\n- Line 225: You seem to be using zero-shot CoT, and I’d cite “Large Language Models are Zero-Shot Reasoners” too\n- Giving the results with DeepSeek R1, it’d be great to see results for other reasoning models like o4-mini to test whether that’s a key difference. I’m not asking for more experiments, but this could be mentioned in future work.", "limitations": "Yes, the paper describes limitations in the conclusion.", "rating": 2, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Conditional on addressing all the weaknesses, I’d increase the score.** \n\nI think this work studies a relevant problem and is useful for the ML community, but there are things that need to be improved for me to recommend its acceptance.\n\n## Strengths\n\n- The paper studies an important problem for safely deploying models and agents, and the experiments can be used to test belief entrenchment in realistic applications.\n- The authors analyze results for different models (both open and closed source, reasoning models and others), different prompt strategies, and different realistic applications (as previous work has focused on synthetic data or simpler problems).\n\n## Weaknesses\n\n### Writing\n\nThe writing in the abstract and introduction could be a lot clearer. For example, in the abstract, “belief entrenchment” is mentioned twice but not defined until line 16. Stylistically, it’d be better to start the Appendix with a section title instead of a figure. Here is a list of typos I found, but there could be more, so please proofread:\n\n- Typo in line 123 “suffers”\n- Typo in line 559 “your”\n- Typo in line 170 “consists”\n- Typo in line 171 with extra space before comma\n- Typo in line 172 with “an” instead of “a”\n- Line 275 needs fixing\n- In Line 237, you need either Appendix B or “in the Appendix”.\n\n### Figures\n\nMost figures in the paper need to be improved. Figures 4, 5, 6, and 7 have tiny fonts, and it’s almost impossible to read them. Figures 4 and 5 could take less space (e.g., Figure 5 can go side by side) to have enough for extra analysis, or other details currently in the Appendix.\n\nFigures 1, 2, and 3A don’t communicate well the problem you’re tackling. For example, Figure 1 is an opportunity to summarize visually your problem, and I think it’s missing the mark. The reasoning steps should be defined as such, and the right side (Weak No, No, Strong No, etc) is not clear where it’s coming from.\n\nThe caption of Figure 5 could include more information (e.g., domain details). Similarly, captions in Figures 1 and 2 should elaborate more.\n\n### Analysis\n\nThe analysis in Table 1 would benefit from reporting statistical significance at different levels (e.g., typically * (p<0.05), ** (p<0.01), *** (p<0.001)) and their effect size. Without that, it’s hard to say whether statistically the effect of belief entrenchment is that relevant (e.g,. large sample size can lead to small p-values even without significant results).\n\nIn Figure 4 and lines 264-266, you claim “We observed that the Martingale Score and Brier score are positive correlated across all setups.” If you look at Figure 4, it seems clear that you’re dealing with a problem of nonlinearity or heteroscedasticity. A few outliers seem to have a lot of leverage on the regression results, as you can clearly see for CoT on OpenReview (red) with that point at (0.15, 0.4). I’d study the leverage of highly influential points and report regression results after that. My impression is that you won’t have meaningful results to claim a positive correlation, except for CoT on Forecasting.", "was_revised": true, "final_justification": "I think the idea in this paper is really interesting, and I was willing to increase my initial score (Borderline Reject) if my concerns were addressed. However, **the rebuttal was not convincing at all, so I'm decreasing my score** (Reject). I replied to their rebuttal comments very early on in the discussion period, and I heard back after 5 days. I've considered their last comments, but I believe the evaluation is flawed.\n\nThere are weaknesses that I believe can be addressed, such as the writing (lots of typos and lack of clarity), figures (no good explanation of the process or very low quality), and reporting statistical effect sizes.\n\nHowever, **I'm most concerned about the analysis.** Looking at Figure 4, it's evident that linear regression might not be adequate in this case. There seems to be nonlinearity and/or heteroscedasticity, which make the numerical results irrelevant.\n\nIn the rebuttal, the authors reported results controlling for outliers, but first did so strategically only for the cases where the issues I mentioned are not concerning. This seems to have been a misunderstanding; in the end, they reported all the results. However, there's only positive correlation in one setup. \n\nThey have changed the outlier removal criteria and included more trajectories because their analysis was underpowered. Moreover, they claim that looking at Debate results and CoT for OpenReview is not very relevant because those datasets are noisy. Coincidentally, there's no positive correlation for those, which means that they're asking the reviewers to disregard the bad results and only consider the good ones.\n\n**I urge all reviewers and AC to review this thoroughly.**", "rebuttal": "Thanks for the review! We highly appreciate the effort and will respond to specific comments.\n\n### Clarifying the statistical significance and effect size of our findings \n\nReporting statistical significance at different levels is a very valid constructive point. We will adopt it in the final publication.\n\nWe thank the reviewer for this important suggestion. To better demonstrate the practical relevance of our findings, we will report the coefficient of determination (R2) as a standardized measure of effect size.\n\nThe Martingale Score (the unstandardized regression coefficient, β1) is itself a direct measure of the effect, but supplementing it with R2 will clarify the proportion of variance in belief updates explained by the prior belief. For instance, in the Gemini 2.0 Flash-NoPrompt-CoT-Forecasting setup, the R2 is 0.0589, indicating the prior belief accounts for approximately 6% of the variance in the update.\n\nMost importantly, we argue that the practical relevance of the Martingale Score is most clearly validated by its **strong correlation with prediction accuracy** (the Brier score), as shown in our Figure 4. This direct link between a higher Martingale Score and a tangible drop in performance confirms its practical importance. We will add the R2 values to our results table and clarify this discussion in the final manuscript.\n\n### Clarifying the correlation between Martingale Score and Brier score \n\nWe thank the reviewer for their sharp observation, as positive correlation does not universally similarly apply to all setups. \n\nRegarding your concern on outliers being overly influential - we conducted three other series of regressions:\n\n- 1. (Light-handed) We remove the 10 most significant outliers (out of 78 points in total) as measured by Cook's distance and re-do all regressions.\n- 2. (Heavy-handed) We remove the 30 most significant outliers as measured by Cook's distance and re-do all regressions.\n- 3. (Conservative): Statistical threshold (4/n Cook's distance)\n\n- The table below summarizes the results, specifically concerning the linear coefficient of the Martingale Score (the dependent variable being Brier score) when under the Debate regime.\n- Ignoring the “+ Reasoning + Interaction” the row given its extremely low statistical power and high p-value, overall, removing outliers does **not** change the sign of the coefficient. This can be confirmed in one case with statistical significance, and another with marginal significance.\n\n*Debate x Martingale Score Interaction Effects Across All Approaches (n=78)*\n\n| Model Specification | No Outlier Handling | Light (≤10) | Heavy (≤30) | Conservative | \n|---------------------|-------------|-------------|-------------|--------------|\n| **+ Reasoning + Interaction** | +0.064 (p=0.861) | +0.409 (p=0.199) | +0.030 (p=0.861) | +0.210 (p=0.565) |\n| **+ Domain + Reasoning + Interaction** | -0.422 (p=0.209) | **-0.547 (p=0.060)** | **-0.472 (p=0.040)** | -0.363 (p=0.239) |\n| **+ Model + Interaction** | -0.481 (p=0.132) | -0.378 (p=0.130) | -0.236 (p=0.179) | -0.252 (p=0.374) |\n| **+ Prompt + Interaction** | -0.193 (p=0.542) | -0.111 (p=0.670) | -0.313 (p=0.133) | -0.195 (p=0.495) |\n\n*Note: Light, heavy, and conservative methods show robust results consistent with no outlier handling.*\n\nThe discrepancy among setups on their martingale-brier scores relations\n\nWhile our results vary across setups, this is expected as the ability to validate the Martingale Score depends on the quality of the experimental conditions.\n\nOur most convincing results come from the setups best suited for this analysis. For example:\n\n- **Chain-of-Thought (CoT) is more trustworthy than Debate** due to a significantly larger sample size.\n\n- **The Forecasting domain is more reliable than OpenReview** because its ground-truth signal is objective and less noisy than the subjective decisions of a few human reviewers. [1]\n\nThe primary challenge is finding data with reliable ground truth that the LLM has not seen during training. Forecasting is the ideal domain for this purpose. The strong results obtained in this domain provide the core validation for the martingale score as a meaningful, unsupervised measure of belief entrenchment. In the following research, we hope to find more domains where martingale scores can meaningfully predict either ground truth labeling or collective voted judgments if there is no ground truth. Two domains we plan to run: OpenReview but to assess the relation between martingale score and citation number (as this more closely aligns with collective voted judgments) and Reddict r/AITA (also collective voted judgments) [2]\n\n### Other suggestions \n\nAlso, we very much appreciate that you spend time identifying typos and making suggestions about writing and figures. **All the suggested changes will be reflected in the final manuscript.**\n\n### Reference \n\n[1] Beygelzimer et al. Has the Machine Learning Review Process Become More Arbitrary as the Field Has Grown? The NeurIPS 2021 Consistency Experiment\n\n[2] Cheng et al. Social Sycophancy: A Broader Understanding of LLM Sycophancy"}]

Author rebuttal:

Meta-review: The paper investigates the "belief entrenchment phenomenon" in Large Language Models (LLMs). This phenomenon refers to a model's tendency to resist updating its beliefs in light of new evidence, instead sticking to its prior beliefs. To assess this phenomenon, the authors introduce an unsupervised, regression-based metric called the Martingale Score. This score quantifies deviations from the Bayesian Martingale property, which posits that belief updates should be unpredictable based on previous beliefs. Instances of such deviations indicate irrational entrenchment. The authors discover that these violations are prevalent across various LLM models, reasoning techniques (including Chain-of-Thought and Debate), and problem domains. Furthermore, they validate the Martingale Score through experiments that demonstrate its ability to predict ground-truth accuracy in domains where valid labels are available.

In total, this paper received five reviews, with four recommending acceptance and one recommending rejection. All reviewers concur on the significance of the study's motivation, the originality of the Martingale Score, and the quality of the paper. However, the statistical significance of belief entrenchment and the scoring metrics have been contentious topics during the rebuttal phase.

In summary, I recommend acceptance of the paper, but I strongly encourage the authors to consider the feedback provided by Reviewer #K3qf. Additionally, improvements could be made in the clarity of the introduction and the readability of the figures.
Decision: Accept (poster)