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[2507.11768] LLMs are Bayesian, In Expectation, Not in Realization
Summary
This paper explores the Bayesian nature of large language models (LLMs) in expectation rather than realization, highlighting the impact of positional encodings on model performance and exchangeability.
Why It Matters
Understanding the Bayesian characteristics of LLMs is crucial for improving their design and performance. This research provides insights into how positional encodings affect model behavior, which can inform future developments in machine learning and AI applications.
Key Takeaways
- Positional encodings in LLMs disrupt exchangeability, affecting Bayesian behavior.
- Performance should be evaluated based on expected outcomes over exchangeable multisets.
- Empirical findings show significant gaps between expectation and realization in LLM outputs.
Statistics > Machine Learning arXiv:2507.11768 (stat) [Submitted on 15 Jul 2025 (v1), last revised 22 Feb 2026 (this version, v2)] Title:LLMs are Bayesian, In Expectation, Not in Realization Authors:Leon Chlon, Zein Khamis, Maggie Chlon, Mahdi El Zein, MarcAntonio M. Awada View a PDF of the paper titled LLMs are Bayesian, In Expectation, Not in Realization, by Leon Chlon and 3 other authors View PDF HTML (experimental) Abstract:Exchangeability-based martingale diagnostics have been used to question Bayesian explanations of transformer in-context learning. We show that these violations are compatible with Bayesian/MDL behavior once we account for a basic architectural fact: positional encodings break exchangeability. Accordingly, the relevant baseline is performance in expectation over orderings of an exchangeable multiset, not performance under every fixed ordering. In a Bernoulli microscope (under explicit regularity assumptions), we bound the permutation-induced dispersion detected by martingale diagnostics (Theorem~3.4) while proving near-optimal expected MDL/compression over permutations (Theorem~3.6). Empirically, black-box next-token log-probabilities from an Azure OpenAI deployment exhibit nonzero expectation--realization gaps that decay with context length (mean 0.74 at $n = 10$ to 0.26 at $n = 50$; 95\% confidence intervals), and permutation averaging reduces order-induced standard deviation with a $k^{-1/2}$ trend (Figure~2). Controlled from-scratch training abla...
