Dutch Books for Language Models
Uses de Finetti theorem and linear programming to quantify incoherence in language model probabilistic forecasts via maximum Dutch-book profit.
Key Findings
Methodology
This study applies de Finetti’s theorem to construct a set of probabilistic forecasts over known logical events, then employs linear programming to compute the maximum guaranteed profit (Dutch-book) an arbitrageur could secure. Data from stock returns is used, where models predict the probability of future returns falling into specific bins. The event space is built based on logical relations among these events, enabling evaluation of coherence without outcome labels. The approach assesses whether model probabilities can be extended to a full probability measure, indicating internal consistency.
Key Results
- Across 100 stock-days, average arbitrage profit was 0.00207, indicating widespread incoherence, with some models reaching profits over 1%. Variations across models spanned 100-fold, showing significant differences in internal consistency. Complex logical dependencies and irrelevant contextual details increased incoherence by an order of magnitude. Models with richer event structures exhibited higher arbitrage profits, confirming that logical complexity correlates with incoherence. Prompt strategies influenced coherence, with additional information sometimes reducing consistency, highlighting sensitivity to elicitation protocols.
- Models' incoherence correlates with accuracy but varies more widely, with arbitrage profits showing larger spread than traditional Brier scores. Experiments with multi-asset and multi-day questions reveal that logical dependencies and cross-event links significantly elevate incoherence, emphasizing the importance of logical reasoning in probabilistic forecasting. The method's robustness is confirmed through ablation studies and diverse prompt manipulations.
- The findings demonstrate that current language models, despite high performance, harbor fundamental probabilistic inconsistencies. This undermines their reliability in decision-critical applications like finance and policy. The approach offers a scalable, label-free way to detect and quantify these flaws, guiding future training strategies to improve logical coherence and robustness in probabilistic reasoning.
Significance
This research introduces a novel, unsupervised framework for evaluating probabilistic coherence in language models, addressing a core limitation of traditional calibration metrics. By quantifying the maximum arbitrage profit, it provides a direct measure of internal logical consistency, crucial for deploying models in high-stakes domains such as finance, risk management, and policy. The method’s independence from outcome labels makes it broadly applicable, enabling large-scale assessment of models’ reasoning capabilities. The insights gained highlight the need for training regimes that emphasize logical coherence, potentially transforming how probabilistic models are developed and validated, ultimately fostering more trustworthy AI systems.
Technical Contribution
The paper pioneers the integration of de Finetti’s theorem with linear programming to assess model coherence, establishing a global, label-free metric based on maximum Dutch-book profit. This approach extends traditional calibration by directly quantifying internal logical consistency, applicable even when outcomes are unknown. The methodology involves constructing event spaces from logical relations, formulating the maximum arbitrage problem as a linear program, and deriving coherence measures from duality principles. Empirical validation across diverse models and event structures demonstrates its effectiveness, providing a new theoretical and practical tool for probabilistic model evaluation.
Novelty
This work is the first to operationalize de Finetti’s theorem for large-scale, label-free coherence assessment of language model predictions. Unlike existing calibration and scoring rules, which rely on observed outcomes, this method evaluates the internal logical consistency of probabilistic forecasts directly. It introduces a global, optimization-based framework that captures subtle incoherence patterns, especially in complex, multi-asset, multi-event scenarios. The approach bridges foundational probability theory with practical model evaluation, offering a novel perspective on model reliability and robustness.
Limitations
- The approach depends on the accurate construction of event spaces; poorly defined events can bias results. Computational complexity increases with event set size, limiting scalability. The method is primarily validated in financial forecasting; applicability to other domains remains to be tested. It assesses internal coherence but does not directly improve model training; integrating it into training pipelines is future work.
Future Work
Future research will explore incorporating coherence constraints into training objectives, possibly via reinforcement learning or adversarial approaches, to produce models with inherently consistent probabilities. Extending the framework to multi-modal and more complex event structures, including causal and temporal dependencies, is a priority. Additionally, developing scalable algorithms for larger event spaces and integrating coherence metrics with calibration and other evaluation tools will enhance practical deployment. The ultimate goal is to create models that are both accurate and logically consistent, suitable for high-stakes decision-making.
AI Executive Summary
In recent years, language models have revolutionized natural language processing, demonstrating remarkable capabilities in tasks from translation to prediction. However, their probabilistic forecasts often suffer from internal inconsistencies, undermining trust in critical applications such as finance and policy. Traditional metrics like calibration and Brier scores measure accuracy but fail to capture whether the predicted probabilities adhere to logical constraints. This gap has motivated researchers to seek methods that directly assess the internal coherence of probabilistic predictions.
This study introduces a novel, unsupervised framework based on de Finetti’s theorem, which states that a set of probabilities is coherent if and only if no arbitrage opportunity exists. By formulating the problem as a linear program, the authors compute the maximum guaranteed profit (Dutch-book) an arbitrageur could secure against the model’s predictions. A zero profit indicates perfect coherence, while higher profits reveal increasing incoherence. The approach is applied to stock return data, where models predict the probability of returns falling into specific bins across multiple assets and days.
Empirical results across 15 models and various event structures reveal widespread incoherence, with average arbitrage profits exceeding 0.2 in some cases. Complex logical dependencies, multi-asset questions, and irrelevant contextual details significantly increase incoherence, highlighting the models’ limitations in reasoning about interconnected events. The findings demonstrate that current language models, despite high accuracy, harbor fundamental probabilistic flaws that could impair decision-making.
The significance of this work lies in providing a scalable, label-free method to evaluate and improve probabilistic reasoning in AI systems. It offers a new lens for understanding model reliability, emphasizing the importance of logical consistency alongside accuracy. Future directions include integrating coherence constraints into training, extending the framework to broader domains, and developing models inherently capable of logical probabilistic reasoning. This research paves the way toward more trustworthy, logically sound AI models capable of supporting high-stakes decisions with greater confidence.
Deep Analysis
Background
近年来，语言模型在自然语言处理和预测任务中取得了巨大突破，尤其在金融、医疗等领域展现出强大潜力。早期代表作如GPT系列、BERT等，主要关注模型的准确率和校准指标，但对其概率预测的逻辑合理性关注不足。传统校准指标（如Brier分数）虽能反映部分性能，但无法检测模型内部的矛盾。近年来，学界逐渐认识到模型在复杂事件推理中的不一致性，特别在多资产、多时间跨度环境下表现突出。本文借助de Finetti定理，提出无监督的概率一致性检测方法，为模型在未观察到的事件上的推理提供理论支撑。
Core Problem
当前，语言模型在概率预测中存在明显的不一致性，表现为套利机会和逻辑矛盾。传统评估指标难以捕捉模型内部的逻辑合理性，尤其在多事件、多资产、多时间跨度的复杂场景中。这限制了模型在实际决策中的可靠性，尤其在金融风险管理、政策制定等关键领域。如何量化模型预测的内在逻辑合理性，成为亟待解决的问题。现有方法多依赖观察到的结果标签，难以应用于未来或未决事件，缺乏普适性和无监督特性。
Innovation
本研究的核心创新在于将de Finetti定理引入概率预测评估，提出线性规划算法计算最大Dutch-book利润，作为模型内在不一致性的量化指标。该方法无需观察到的结果标签，适用于未决事件，具有普适性。通过系统性实验，验证了模型在复杂逻辑关系、多资产、多时间跨度下的预测不一致性，揭示了模型在推理中的潜在漏洞。创新点还在于提出全局性、无监督的概率合理性检测框架，为未来模型训练提供理论基础和实践工具。
Methodology
- �� 构建事件空间：利用股票收益数据，定义未来收益落在特定区间的事件，形成逻辑关系明确的事件集合。• 事件的线性表示：通过事件的交并补关系，建立事件的代数结构，生成事件的原子空间。• 预测概率向量：模型输出每个事件的预测概率，形成概率向量p。• 线性规划：定义最大套利利润问题，目标是最大化套利者在所有事件上的最小净收益，约束为总下注额度。• 计算套利利润：利用对偶理论，将最大套利利润转化为对事件空间的距离度量。• 评估模型：在不同模型、事件组合和提示策略下，计算套利利润，分析模型的概率一致性。
Experiments
采用CRSP股票收益数据和Refinitiv新闻，构建100个股票日样本，评估15个模型在不同事件集上的预测。设计多资产、多天、多事件组合，比较不同提示策略对模型一致性的影响。通过线性规划求解最大套利利润，采用bootstrap方法计算95%置信区间。还进行了逻辑关系丰富度、上下文干扰和多次重复等消融实验，验证方法的稳健性和敏感性。
Results
模型的平均套利利润为0.00207，显著高于随机或完美一致的水平，表明普遍存在预测不合理性。不同模型套利利润差异达100倍，说明模型内部逻辑一致性差异巨大。丰富的逻辑关系和无关上下文会显著增加套利利润，提示模型在复杂推理中表现不足。多资产、多天、多事件组合的套利利润更高，验证了逻辑依赖性对预测合理性的影响。模型在不同提示策略下表现差异明显，提示训练中应关注逻辑一致性。
Applications
该方法可用于金融、保险、风险评估等领域，帮助识别模型潜在的逻辑漏洞，提升模型可信度。未来可结合训练策略，增强模型在复杂推理中的概率合理性，推动模型在决策支持中的应用。无监督检测机制也适合大规模模型评估和模型改进，促进模型的稳健性和可靠性。
Limitations & Outlook
方法依赖事件空间的合理定义，事件设计不当可能影响评估效果。计算复杂度随事件集扩大而增加，限制大规模应用。当前主要在金融场景验证，泛化到其他领域需进一步验证。模型在极端或未见事件上的表现仍需关注，未来需结合训练优化提升模型逻辑一致性。
Plain Language Accessible to non-experts
想象你在一家工厂里，工厂的任务是生产各种不同的产品。每个产品都需要按照一定的规则和逻辑来生产，比如某个零件必须先装好，再装配到最终产品中。工厂的管理者希望每个工人都能严格按照规则操作，否则会出现问题。现在，假设我们用一个智能机器人来预测工厂里每个环节的成功概率。这个机器人告诉你，某个零件装配成功的概率是70%，另一个是80%。但如果这个机器人说，两个零件都成功的概率是60%，而它又说第一个成功的概率是70%，第二个是80%，这就不合理了——因为两个事件的概率关系必须符合一定的逻辑。我们用一种特殊的方法，像工厂检查员一样，看看这个机器人给出的概率是否符合逻辑。我们会模拟各种可能的情况，看看机器人是否总会“赚”到钱（即出现漏洞）。如果它总能“赚”到钱，说明它的预测不合理，就像你在游戏中用漏洞赢了别人一样。这个方法不用知道真正的答案，只是检查预测的合理性。通过这个过程，我们可以发现很多模型在复杂的逻辑关系中会出现预测不一致的问题，就像工厂里的工人偶尔会违反操作规则一样。这提醒我们，未来的模型需要更好地理解和遵守逻辑关系，才能在实际应用中更可靠。
ELI14 Explained like you're 14
想象你在玩一个超级复杂的游戏，你需要猜测未来会发生什么，比如明天的天气、考试的难度或朋友的心情。现在，有个聪明的机器人也在帮你猜，但有时候它的答案会互相矛盾，就像你说“今天晴天”，但又说“明天会下雨”，这两件事怎么可能都是真的呢？这就像你在猜谜时，答案之间没有逻辑联系，导致你可以用一些小技巧，总是赢得游戏。科学家们用一种叫“de Finetti定理”的数学方法，来检查这些猜测是否合理。具体做法是：他们让机器人猜出一堆事件的概率，然后模拟各种可能的结果，看看机器人是不是总能“赚到钱”——也就是说，它的猜测是否有漏洞。如果能“赚到钱”，说明它的猜测不合理，就像你在游戏中用漏洞赢了别人一样。这个方法不用知道真正的答案，只是看猜测是否符合逻辑。研究发现，很多模型在面对复杂问题时，预测会出现很多不一致的地方，就像游戏里出现的漏洞一样。这告诉我们，未来要让机器人更聪明，不仅要让它猜得准，还要让它的猜测符合逻辑，就像我们在玩游戏时要遵守规则一样。这样，机器人才能在现实生活中帮我们做出更靠谱的决定，比如投资、医疗或天气预报。
Abstract
People increasingly use language models to support life decisions. Many such decisions involve a probabilistic forecast: How likely is a major life event, a natural disaster, or an economic outcome? Users of language models may implicitly trust that these forecasts fall out of a coherent world model. In this paper, we evaluate the coherence of language model probabilistic forecasts through a procedure that builds on a theorem due to de Finetti. We elicit forecasts from language models across events generated from stock returns data. We then use linear programs to compute the largest Dutch-book profit - the profit an arbitrageur could guarantee by betting against model-generated probabilities - which we use as a measure of incoherence. Our procedure does not require outcome labels, so we can evaluate coherence even in settings where outcomes are not observed or have not yet resolved. We find substantial evidence of incoherence in language model forecasts. Such incoherence increases when there are richer logical relationships between events, and irrelevant contextual details can increase incoherence by an order of magnitude. We conclude by discussing how alternative training strategies may improve probabilistic coherence.