Title: Martingale Score: An Unsupervised Metric for Bayesian Rationality in LLM Reasoning
Venue: NeurIPS 2025
Accepted: yes
Keywords: Martingale Score

Martingale property

bayesian reasoning

LLM

unsupervised

evaluation

belief entrenchment

forecasting
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

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.

1 Introduction

Consider a stubborn person who refuses to be shown wrong, or a king surrounded by sycophants. Human beings, in their efforts to seek true beliefs, often end up entrenching their preexisting beliefs, whether due to their own confirmation bias [Klayman, 1995] or due to external

30% Weak No

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Should the state of New York (NY) be split into two states?

Polls show limited enthusiasm.

Splitting causes administrative headache.

California repeated proposals have failed.

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., 2024, 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 Imhoff, 2023].

Having learned patterns of human reasoning from human-generated text data at an Internet scale, large language models (LLMs) are designed to help humans by answering their questions [Luo et al., 2022] or assisting with their tasks [Zheng et al., 2023]. Following their initial success in sophisticated cognitive tasks [Brown et al., 2020, Zhao et al., 2023], a range of reasoning techniques have emerged based on these models. From the earliest Chain-of-Thought (CoT) technique [Wei et al., 2022] to the more recent paradigm of inference-time scaling via reinforcement learning [Zelikman et al., 2024, Jaech et al., 2024, Guo et al., 2025] ( reinforced reasoning henceforth), these methods aim to help language models in truth-seeking , i.e., gaining a correct understanding via argumentation, evidence-seeking, and trial-and-error at inference time.

However, such reasoning in language models can deviate from truth-seeking due to belief entrenchment -asystematic tendency to update beliefs in favor of prior opinions rather than in response to new evidence (Figure 1). When this occurs, it can degrade the accuracy of a model's 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 et al., 2024a]. Although recent studies suggest the presence of belief entrenchment in LLMs [Schmidgall et al., 2024, Shi et al., 2024, Sumita et al., 2024], demonstrating it rigorously remains challenging. In any single example, it is difficult to distinguish a justified, prior-consistent update (e.g., a prediction later validated by evidence) from a biased update (e.g., representing evidence to maintain an unfalsifiable prior) [Atallah et al., 2021, 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 tendency toward belief entrenchment. Specifically, we define belief entrenchment as a violation of the Martingale property -acore 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 as a principled and unsupervised indicator of irrational belief entrenchment. Concretely, if a model's future belief can be reliably predicted from its prior belief across reasoning iterations-quantified via the goodness-of-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 forecasting, value-laden questions, and academic paper review (hereafter, Forecasting, r/ChangeMyView, and OpenReview, respectively). Across a wide range of models (e.g., GPT-4o, Llama 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, a 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 an 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/ChangeMyView, and OpenReview), prompts (priorconforming, no-prompt, critical-thinking), and model families (GPT, DeepSeek, Gemini, Llama, etc).

Connecting Belief Entrenchment to Accuracy Loss . We find that belief entrenchment consistently predicts an accuracy drop in problem domains where ground truth labels exist (e.g., forecasting). This suggests that the Martingale Score serves as a 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 theorem. Gupta et 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 beliefs 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 in 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., 2025a], 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 as they are trained on a large corpus of human data [Echterhoff et al., 2024]. 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) [Ye et al., 2024], and certainty bias [Zhou 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., 2021]. It is also hypothesized that most cognitive biases are variants of confirmation bias [Oeberst and Imhoff, 2023]. 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 AI 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 et al., 2023], and a trade-off between truthfulness and utility [Su et al., 2025]. Honest AI emphasizes that AI does not 'intentionally' assert anything that does not align with its beliefs in pursuit of rewards [Evans et al., 2021]. This is closely related to the concept of deception in LLMs [Hubinger et al., 2024, Su et al., 2025]. In contrast, we are interested in a less investigated concept: truth-seeking AI [Koralus, 2025], the process of weighing evidence and discovering novel findings. Truth-seeking may become an important objective of LLMs as they are 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 AI explicitly aims for gaining truth, but 'truth-seeking' may not be a single metric to 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 to 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 and a measure of one's prior beliefs [Berthet, 2021]. This is a significant challenge by itself.

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, 2021]. In other words, under the Martingale property of Bayesian statistics, reasoners should never make predictable belief updates. Different from confirmation bias, because of how belief entrenchment is measured, it is defined as a statistical property, rather than an individual tendency.

Belief 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 is LLMs' expressed 'beliefs' through their output - when LLMs are entrenched by their own 'beliefs,' they express high confidence in their own stated output, unjustified sense of confidence would mislead LLM users.

Belief update Relatedly, belief update refers to the change in confidence in LLM's output. And we use LLM judge to assess confidence, as detailed in Section 5.1. We treat the process of having LLMs do chain-of-thought reasoning as a 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, is 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-AI interactions.

Bayesian Reasoning and Truth-Seeking Previous research presents mixed results to the question whether LLM in-context learning is Bayesian [Gupta et al., 2025, Falck et al., 2024]. Meanwhile, specific reasoning failure instances were reported: sycophancy [Sharma et al., 2023], 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 al., 2024]. Reasoning evaluation is usually conducted by measuring ground truth accuracy [Sawada et al., 2023, Phan et al., 2025]. Such outcome-based evaluation falls short of our expectations because it does not tell us how LLM reasoning achieves superior performance, hence whether it would generalise out-of-distribution. 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-solvers need to take contextual information into account. We need process-based metrics to answer the question of 'how' [Mondorf and Plank, 2024a].

Misleading Human-AI Interactions Research on sycophancy [Oeberst and Imhoff, 2023], and conformity [Weng et al., 2025] demonstrates that LLMs 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, it 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 are trained on data containing human beliefs) may lead to lock-in of false beliefs collectively [Burton et al., 2024, Qiu et al., 2025b, Weidinger et al., 2023].

4 Measuring Belief Entrenchment with Martingale Score

To address these problems, we propose the Martingale Score as a principled, unsupervised measure of belief entrenchment in LLM reasoning. Effective reasoning requires the capacity to update beliefs in response to new evidence, a 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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Martingale Score calculates the statistics of predictability of belief update solely based on prior.

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4.1 Defining the Martingale Score

To compute the Martingale Score, we perform the regression ∆ b = β 1 · b prior + β 0 + ϵ , where b prior are the prior probabilities, ∆ b = b posterior -b prior, and ϵ is the error term.

We define the sample estimate ˆ β 1 of the linear coefficient as the Martingale Score M , with the Ordinary Least Squares (OLS) method. Equivalently, when there are n samples,

Martingale Score M measures the extent to which the prior belief b prior positively (or negatively, if M < 0 ) predicts belief update ∆ b . Using OLS allows us to test the statistical significance of M , assessing whether the relationship between ∆ b and b prior is distinguishable from zero (e.g., via a t-test with p < 0 . 05 ).

Wechoose the M = ˆ β 1 definition for its simplicity, lack of confounders (as opposed to R 2 (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 of update, which neglects the magnitude of belief updates and produces random-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,

This implies that the direction of a Bayesian agent's belief update (whether positive or negative) should not be predictable from the prior alone. Indeed, the Martingale property has been shown to be the defining characteristic of Bayesian rationality [Molavi, 2021].

The Martingale property (2) implies that the prior b prior is statistically exogenous to the belief update ∆ b [Hayashi, 2011]. This exogeneity yields two desirable properties: the expected coefficient E[ ˆ β 1 ] = 0 in the regression (1), and consistency of the estimator, i.e., ˆ β 1 → 0 in probability as the number of samples approaches + ∞ [Hayashi, 2011]. These properties justify the Martingale Score , defined in (1), as a principled measure of violations of the Martingale property (2). Formally,

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Should the state of New York (NY) be split into two states?

Splitting it might dilute NY's rich culture identity.

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Splitting causes administrative headache.

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California repeated proposals have failed.

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Chain of Thought Reasoning

(a) An example of real reasoning trajectory

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Proposition 1 If the Martingale property holds, the population coefficient β 1 is 0 and the sample estimate ˆ β 1 is an unbiased and consistent estimator of β 1 . In this specific case where β 1 = 0 , this implies E ( M ) = 0 and M p - → 0 , as n →∞ . 3

As the Martingale property is a prerequisite for Bayesian reasoning (i.e., ensuring that updates are 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.

5 Experiment Setups

We set up experiments to 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

LLMs are often poorly calibrated: their confidence scores-whether expressed through token probability or self-reported scores- do not reliably reflect their true degree of belief [Pawitan and Holmes, 2024, Zhou et al., 2023]. To address this, we adopt an 'expressed belief' approach rather than relying on 'internal belief,' which remains an open research problem[Hase 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 a belief score b ∈ [0 , 1] . To ensure robustness, we evaluate

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 reasoning-specifically, how models incorporate new evidence during the reasoning process-we select domains that meet the following criteria:

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 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 a Bayesian reasoner would use to revise its beliefs. This allows us 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 to 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: (1) forecasting questions come with ground truth labels once resolved; and (2) achieving high accuracy on a set of forecasting questions reflects Bayesian-like reasoning, as 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 a 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/ChangeMyView subreddit [Tan et al., 2016]. These discussions are explicitly designed to explore whether individuals (or in this case, LLMs) can revise their opinions when presented with counterarguments. 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 entrenchment. 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 a 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

We 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 a topic and participate in debate, allowing a less informed observer to arrive at a better understanding of the subject.

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: a prior-conforming prompt and a critical-thinking prompt.

The prior-conforming prompt reinforces a belief aligned with the model's likely initial stance, serving both as a sense check and as a potential lower bound for belief entrenchment and accuracy. In contrast, the critical-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 our empirical findings. We first establish an association between belief entrenchment (as measured by the Martingale Score) and drops in ground-truth accuracy, as a vindication for the former. We then benchmark a range of closed- and open-weights models on 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 1 shows the Martingale Score of different models under different experiment setups. A positive Martingale Score M indicates that there is an M -unit increase in ∆ b per unit increase in b prior. 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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Notably, we observe overall more severe belief entrenchment in value-laden domains such as r/ChangeMyView (when using the same model, under the same system prompt, the Martingale Score is always higher in r/ChangeMyView than in Forecasting), suggesting that belief entrenchment might be a more serious concern in domains where we cannot evaluate LLM performance against ground truth. We also intended to manipulate the degree of belief entrenchment by using a critical thinking prompt and a prior-conforming prompt. We observe that belief entrenchment is more severe under a prior-conforming prompt and less severe under a critical thinking prompt in Forecasting, but not in r/ChangeMyView 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 is meant to happen, as LLMs are instructed to emphasize arguments in favor of their prior belief, and such reasoning harms performance. However, we noticed that even under 'no system prompt' and 'critical thinking prompt', belief entrenchment also happens, although to a lesser extent ( M Prior-conforming = 0 . 082 ± 0 . 018 , M No-prompt = 0 . 075 ± 0 . 014 , M Critical-thinking = 0 . 072 ± 0 . 018 , with 95% CI). The results are significant, demonstrating a consistent tendency of belief entrenchment in LLM behavior.

Belief Entrenchment Harms Accuracy As the ultimate aim of reasoning in 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 4 shows 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 of one setup (of one model, with one type of system prompt and one reasoning mode, on > 100 questions in one problem domain.

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While outcome-based metrics (such as Brier score) measure model performance, they reveal little about why models achieve high performance, and for this reason, we need process-based metrics on reasoning [Mondorf and Plank, 2024b]. We show a 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., r/ChangeMyView) as well. Table 1 demonstrates that, at least with CoT, belief entrenchment consistently happens regardless of prompt types and models. The comparable level of belief entrenchment ( M CMV-CoT = 0 . 103 ± 0 . 013 , M Forecasting-CoT = 0 . 037 ± 0 . 011 , M OpenReview-CoT = 0 . 086 ± 0 . 012 , with 95% CI) corresponds to accuracy drops in forecasting and OpenReview (as in Figure 4 and Figure 5), where ground truth exists, suggesting consistently worsened judgment under belief entrenchment. Ruling out potential artifacts (presented in Section 6.1), we thus think the Martingale Score is a 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-LLM agreement analysis, all judges (LLMs or humans) show a strong correlation with GPT-4o.

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 to 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 human-LLM judges (e.g., human evaluator 1 VS GPT-4o). Full results can be seen in Table 2.

All judges show a large positive correlation with GPT-4o, and all results are statistically significant ( p < 0 . 001 ) 4 .

Considering results from both cross-judge consistency analysis and human-evaluation validation, it is highly unlikely that our belief entrenchment results stem from judge bias.

7 Conclusion

In this paper, we propose the Martingale Score as a principled and unsupervised measure of belief entrenchment in LLM reasoning. We show, as validation, that the Martingale Score predicts the ground-truth accuracy of a reasoning process, and then apply it to a 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 (belief entrenchment) 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 community-voted decisions can be noisy [Beygelzimer et al., 2023]. However, Table 1 shows that belief entrenchment is a more severe phenomenon in domains where human judgments are required (such as r/ChangeMyView 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 [Guo 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 to 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-ended domains by demonstrating other downstream consequences of belief entrenchment.

To demonstrate the usefulness of Martingale Score as a general-purpose process-based reasoning evaluation, one could extend our pipeline into the evaluation of sycophancy [Sharma et al., 2023],

group conformity [Weng et al., 2025], and inverse scaling [Gema et al., 2025]. Martingale Score as an evaluation metric can be converted into a training objective, as a further test of its robustness and applicability (if in a domain where martingale training gets both Martingale Score and brier score reduced, then it provides new evidence that Martingale Score can evaluate reasoning, and is applicable beyond forecasting). Applications of the Martingale Score in measuring and mitigating belief entrenchment or even polarization [Lefebvre et al., 2024] within human-AI systems (e.g., recommender systems) are another direction for future exploration.

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A Proof of Proposition (1)

To prove this proposition, we need two parts. First, we show that the Martingale property (the ideal Bayesian state), the population coefficient β 1 = 0 . Second, we show that our estimation tool (OLS estimator M ) to measure β 1 is reliable.

A.1 PART 1: Martingale Property implies β 1 = 0

We define β 1 as the population coefficient in the linear regression:

We define the population coefficient β 1 from the linear model (3) as:

by showing that the Martingale property (2) implies

We prove

β

1

= 0

The covariance is defined as:

We show that both terms on the right-hand side containing ∆ b are equal to 0.

First, by the law of total expectation (LTE) and the Martingale property (2):

Second, using LTE and the "taking out what is known" property ( E [ XY | X ] = X · E [ Y | X ] ):

Substituting the Martingale property (2) again:

Substituting these results back into the covariance equation (5):

̸

Therefore, β 1 = 0 / Var( b prior ) = 0 (assuming Var( b prior ) = 0 ).

A.2 PART 2: OLS estimator M is a reliable (unbiased and consistent) estimator for β 1 .

Now we have a true population coefficient β 1 = 0 when LLM is Bayesian (Martingale property); and we need to prove that the OLS estimator ˆ β 1 is a reliable estimator for the true population coefficient β 1 . To be a reliable estimator, M should be unbiased and consistent .

Proof of 'Unbiased' An estimator being unbiased would require ϵ to be uncorrelated with regressor b prior. Starting with the regressor model (3), both sides take the conditional expectation w.r.t b prior :

Since b prior , β 0 , and β 1 are known, they come out from the conditional, hence we get

Now, substitute what we know:

From Martingale property: E [∆ b | b prior ] = 0 ; and from PART 1 (A.1) we know that β 1 = 0 and E [∆ b ] = 0 ; and substitute regression equation (3) again for ∆ b , we know that

Again, in PART 1 (A.1) we proved that β 1 = 0 ; and by definition of OLS we know E [ ϵ ] = 0 , hence we get Eβ 0 = 0 .

Cov(∆

b, b prior

) = 0

.

Substitute in (6), we get:

which simplifies directly to

This is a 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 (1) is the best linear unbiased estimator for β 1 .

Proof of Consistency: An estimator is consistent if it is unbiased (which we proved) and its variance approaches 0 as n →∞ . The variance of the OLS estimator M is:

where σ 2 is the constant variance of the error term ϵ . As the sample size n →∞ , the denominator (the sum of squared deviations) grows to infinity, assuming Var( b prior ) = 0 . Therefore:

Since M is unbiased and its variance converges to 0, it is a consistent estimator.

Unbiased: E [ M ] = β 1 . On average, our Martingale Score will hit the true coefficient.

Consistent: M p - → β 1 . As we add more samples, variance shrinks to 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 β 1 .

Thus, a statistically significant, non-zero Martingale Score M indicates a 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 is the domain that suffers least from belief entrenchment, while OpenReview suffers the most; that the use of debate mitigates belief entrenchment; and that DeepSeek R1 shows exceptional resistance to belief entrenchment, while all other models are comparable to each other. We also conduct limited analysis on the GPT-4o sycophancy incident [OpenAI, 2025] by testing the model in question (Table 3).

We conduct regression analysis to identify the factors that contribute to belief entrenchment as measured by the Martingale Score. Consider the regression formula

where c = ( c domain , c reasoning technique , c model , c prompt ) is a vector of categorical variables, and f 1 , f 2 are linear functions.

We find the best-fit ˆ f 1 with OLS, and use its linear coefficients on different factors as a measure of their contribution to belief entrenchment. Figure 7(a) shows these coefficients.

Problem Domain We see a statistically significant difference between the three problem domains of Forecasting, r/ChangeMyView, and OpenReview acceptance prediction, in increasing order of propensity for belief entrenchment. Since Forecasting is a fact-based domain, while ChangeMyView and OpenReview rely heavily on subjective judgments, the gap in their propensity for belief entrenchment hints, more generally, at 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 a bimodal pattern in its δb distribution - see Figure 6 (a2).

̸

(a1)

(a2)

(b1)

(b2)

(a3)

(b3)

Model Most of the models that we tested, including GPT-4o, Claude 3.5 Haiku, Gemini 2.0 Flash, DeepSeek V3, Llama 4 Maverick, and Llama 4 Scout, show comparable levels of propensity for belief entrenchment, with only small and statistically insignificant differences. The only outlier is DeepSeek R1, which exhibits a significantly lower tendency for belief entrenchment compared to all other models. This observation is in line with Figure 7(b), where it is shown that the belief updates made by DeepSeek-R1 are more likely to point toward the ground truth compared to the average of other tested models.

System Prompt We compare three choices of the system prompt: a prior-conforming one, a critical one, and omitting it altogether ( none ). We 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. A 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 a critical thinking-focused system prompt, a lot can still be lost when a bad, prior-conforming system prompt is put in place. According to OpenAI [2025], a similar cause is partially responsible for the April 2025 sycophancy incident in GPT-4o. 5

(a)

(b)

C Implementation Details

Our code and data can be found in 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 NO back-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

Prompt for Chain-of-Thought

Prompt for Debate

Question Construction for the OpenReview Domain

C.3 Hyperparameters and Compute Resources

This study is carried out entirely with API-based inference, with a total cost of 1,500 USD.

During inference, we use a temperature of 0 . 1 for models under evaluation, 0 . 3 for belief measurement. The only exception is Gemini 2.0 Flash, with which we use a temperature of 1 . 0 to avoid RECITATION errors.

NeurIPS Paper Checklist

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Please provide a short (1-2 sentence) justification right after your answer (even for NA).

The checklist answers are an integral part of your paper submission. They are visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will be asked to also include it (after eventual revisions) with the final version of your paper, and its final version will be published with the paper.

The reviewers of your paper will be asked to use the checklist as one of the factors in their evaluation. While "[Yes] " is generally preferable to "[No] ", it is perfectly acceptable to answer "[No] " provided a proper justification is given (e.g., "error bars are not reported because it would be too computationally expensive" or "we were unable to find the license for the dataset we used"). In general, answering "[No] " or "[NA] " is not grounds for rejection. While the questions are phrased in a binary way, we acknowledge that the true answer is often more nuanced, so please just use your best judgment and write a justification to elaborate. All supporting evidence can appear either in the main paper or the supplemental material, provided in the appendix. If you answer [Yes] to a question, in the justification please point to the section(s) where related material for the question can be found.

IMPORTANT, please:

Delete this instruction block, but keep the section heading 'NeurIPS Paper Checklist" ,

Keep the checklist subsection headings, questions/answers and guidelines below.

Do not modify the questions and only use the provided macros for your answers .

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: Two main claims (that belief entrenchment is widespread and that it predicts accuracy) are demonstrated in the Results section.

Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

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Guidelines:

The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.

The authors are encouraged to create a separate "Limitations" section in their paper.

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Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn 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.\n\nSTRENGTHS AND WEAKNESSES:\n**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.\n\nQUESTIONS:\n- 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.\n\nLIMITATIONS:\nYes, the paper describes limitations in the conclusion.", "TITLE": "Official Review", "RECOMMENDATION": 2, "REVIEWER_CONFIDENCE": 5, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-15"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS AND WEAKNESSES:\n**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.\n\nQUESTIONS:\nPlease refer to the Weakness section for more details.\n\nLIMITATIONS:\nPlease refer to the Weakness section for more details.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-20"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\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.\n\nQUESTIONS:\nQuestions:\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.\n\nLIMITATIONS:\nThe 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.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\n### 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**.\n\nSTRENGTHS AND WEAKNESSES:\n### 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.\n\nQUESTIONS:\n1. 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?\n\nLIMITATIONS:\nThe paper evaluates whether belief entrenchment exists in LLMs, but it does not explore why LLMs exhibit this behavior or what mechanisms drive it.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\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.\n\nQUESTIONS:\nThe 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?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-03"}