LLMs are Bayesian, in Expectation, not in Realization

• URL: http://arxiv.org/abs/2507.11768v1
• Date: Tue, 15 Jul 2025 22:20:11 GMT
• Title: LLMs are Bayesian, in Expectation, not in Realization
• Authors: Leon Chlon, Sarah Rashidi, Zein Khamis, MarcAntonio M. Awada,
• Abstract summary: Large language models adapt to new tasks without parameter updates.<n>Recent empirical findings reveal a fundamental contradiction: transformers systematically violate the martingale property.<n>This violation challenges the theoretical foundations underlying uncertainty quantification in critical applications.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Large language models demonstrate remarkable in-context learning capabilities, adapting to new tasks without parameter updates. While this phenomenon has been successfully modeled as implicit Bayesian inference, recent empirical findings reveal a fundamental contradiction: transformers systematically violate the martingale property, a cornerstone requirement of Bayesian updating on exchangeable data. This violation challenges the theoretical foundations underlying uncertainty quantification in critical applications. Our theoretical analysis establishes four key results: (1) positional encodings induce martingale violations of order $\Theta(\log n / n)$; (2) transformers achieve information-theoretic optimality with excess risk $O(n^{-1/2})$ in expectation over orderings; (3) the implicit posterior representation converges to the true Bayesian posterior in the space of sufficient statistics; and (4) we derive the optimal chain-of-thought length as $k^* = \Theta(\sqrt{n}\log(1/\varepsilon))$ with explicit constants, providing a principled approach to reduce inference costs while maintaining performance. Empirical validation on GPT-3 confirms predictions (1)-(3), with transformers reaching 99\% of theoretical entropy limits within 20 examples. Our framework provides practical methods for extracting calibrated uncertainty estimates from position-aware architectures and optimizing computational efficiency in deployment.

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