# Martingale Score: An Unsupervised Metric for Bayesian Rationality in LLM Reasoning

arXiv: https://arxiv.org/abs/2512.02914 · HTML: https://mlsanity.com/paper/2512.02914/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 37 of its 71 references that are in the map.
- First posted on arXiv 2 Dec 2025.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)

## Builds on (most cited first, 30 of 37)

- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [TruthfulQA: Measuring How Models Mimic Human Falsehoods](https://mlsanity.com/paper/2109.07958/) (2021)
- [A Survey of Large Language Models](https://mlsanity.com/paper/2303.18223/) (2023)
- [MathVista: Evaluating Mathematical Reasoning of Foundation Models in Visual Contexts](https://mlsanity.com/paper/2310.02255/) (2023)
- [OpenAI o1 System Card](https://mlsanity.com/paper/2412.16720/) (2024)
- [Towards Understanding Sycophancy in Language Models](https://mlsanity.com/paper/2310.13548/) (2023)
- [Discovering Latent Knowledge in Language Models Without Supervision](https://mlsanity.com/paper/2212.03827/) (2022)
- [Sleeper Agents: Training Deceptive LLMs that Persist Through Safety Training](https://mlsanity.com/paper/2401.05566/) (2024)
- [Humanity's Last Exam](https://mlsanity.com/paper/2501.14249/) (2025)
- [Justice or Prejudice? Quantifying Biases in LLM-as-a-Judge](https://mlsanity.com/paper/2410.02736/) (2024)
- [CRUXEval: A Benchmark for Code Reasoning, Understanding and Execution](https://mlsanity.com/paper/2401.03065/) (2024)
- [Debating with More Persuasive LLMs Leads to More Truthful Answers](https://mlsanity.com/paper/2402.06782/) (2024)
- [Evaluating the Moral Beliefs Encoded in LLMs](https://mlsanity.com/paper/2307.14324/) (2023)
- [Sociotechnical Safety Evaluation of Generative AI Systems](https://mlsanity.com/paper/2310.11986/) (2023)
- [Truthful AI: Developing and governing AI that does not lie](https://mlsanity.com/paper/2110.06674/) (2021)
- [Approaching Human-Level Forecasting with Language Models](https://mlsanity.com/paper/2402.18563/) (2024)
- [Relying on the Unreliable: The Impact of Language Models' Reluctance to Express Uncertainty](https://mlsanity.com/paper/2401.06730/) (2024)
- [Cognitive Bias in Decision-Making with LLMs](https://mlsanity.com/paper/2403.00811/) (2024)
- [Beyond Accuracy: Evaluating the Reasoning Behavior of Large Language Models -- A Survey](https://mlsanity.com/paper/2404.01869/) (2024)
- [Do Language Models Have Beliefs? Methods for Detecting, Updating, and Visualizing Model Beliefs](https://mlsanity.com/paper/2111.13654/) (2021)
- [ARB: Advanced Reasoning Benchmark for Large Language Models](https://mlsanity.com/paper/2307.13692/) (2023)
- [Is In-Context Learning in Large Language Models Bayesian? A Martingale Perspective](https://mlsanity.com/paper/2406.00793/) (2024)
- [When "A Helpful Assistant" Is Not Really Helpful: Personas in System Prompts Do Not Improve Performances of Large Language Models](https://mlsanity.com/paper/2311.10054/) (2023)
- [Do as We Do, Not as You Think: the Conformity of Large Language Models](https://mlsanity.com/paper/2501.13381/) (2025)
- [Inverse Scaling in Test-Time Compute](https://mlsanity.com/paper/2507.14417/) (2025)
- [Bayesian Teaching Enables Probabilistic Reasoning in Large Language Models](https://mlsanity.com/paper/2503.17523/) (2025)
- [Has the Machine Learning Review Process Become More Arbitrary as the Field Has Grown? The NeurIPS 2021 Consistency Experiment](https://mlsanity.com/paper/2306.03262/) (2023)
- [Enough Coin Flips Can Make LLMs Act Bayesian](https://mlsanity.com/paper/2503.04722/) (2025)
- [Cognitive Biases in Large Language Models: A Survey and Mitigation Experiments](https://mlsanity.com/paper/2412.00323/) (2024)
- [Conceptual and Unbiased Reasoning in Language Models](https://mlsanity.com/paper/2404.00205/) (2024)

## Built on by (most cited first, 5 of 5)

- [Accommodation and Epistemic Vigilance: A Pragmatic Account of Why LLMs Fail to Challenge Harmful Beliefs](https://mlsanity.com/paper/2601.04435/) (2026)
- [SWAY: A Counterfactual Computational Linguistic Approach to Measuring and Mitigating Sycophancy](https://mlsanity.com/paper/2604.02423/) (2026)
- [Contextual Drag: How Errors in the Context Affect LLM Reasoning](https://mlsanity.com/paper/2602.04288/) (2026)
- [Illusions of Confidence? Diagnosing LLM Truthfulness via Neighborhood Consistency](https://mlsanity.com/paper/2601.05905/) (2026)
- [Bayesian Intelligence from the Outside](https://mlsanity.com/paper/2609.14724/) (2026)

## Abstract (arXiv)

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 and confirmation bias, 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 are unpredictable from the current belief. We propose the unsupervised, regression-based Martingale Score to measure violations of this property, which signal 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 find such violations to be widespread across models and setups, where the current belief positively predicts future belief updates, a phenomenon which we term belief entrenchment. We identify the models, reasoning techniques, and domains 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 a reasoning process.