The {\alpha}-Law of Observable Belief Revision in Large Language Model Inference

2026-02-26

AI Summary

This study investigates the intrinsic structure of probability distribution shifts in large language models during reasoning, induced by mechanisms such as chain-of-thought prompting, self-refinement, and retrieval augmentation. Through large-scale observation of candidate-answer-level probabilities, regression analysis, and cross-model and cross-dataset generalization tests, the work presents the first empirical evidence that, under diverse reasoning prompt strategies, changes in model output probabilities consistently follow an approximate log-odds linear relationship. This structural regularity achieves a stable fit with an average R² ≈ 0.76 across 4,975 questions and approximately 130,000 observations, revealing a unified mechanism linking evidential signals to probability updates. The findings offer a prompt-sensitive perspective on model calibration and uncertainty propagation, advancing our understanding of how reasoning interventions shape probabilistic beliefs in language models.

Abstract

Large language models (LLMs) that iteratively revise their outputs through mechanisms such as chain-of-thought reasoning, self-reflection, or multi-agent debate lack principled guarantees regarding the stability of their probability updates. We identify a consistent multiplicative scaling law that governs how instruction-tuned LLMs revise probability assignments over candidate answers, expressed as a belief revision exponent that controls how prior beliefs and verification evidence are combined during updates. We show theoretically that values of the exponent below one are necessary and sufficient for asymptotic stability under repeated revision. Empirical evaluation across 4,975 problems spanning graduate-level benchmarks (GPQA Diamond, TheoremQA, MMLU-Pro, and ARC-Challenge) and multiple model families (GPT-5.2 and Claude Sonnet 4) reveals near-Bayesian update behavior, with models operating slightly above the stability boundary in single-step revisions. However, multi-step experiments demonstrate that the exponent decreases over successive revisions, producing contractive long-run dynamics consistent with theoretical stability predictions. Token-level validation using Llama-3.3-70B further confirms similar behavior across both log-probability measurements and self-reported confidence elicitation. Analysis of update components exposes architecture-specific trust-ratio patterns, with GPT-5.2 showing balanced weighting between prior and evidence, while Claude modestly favors new evidence. This work characterizes observable inference-time update behavior rather than internal Bayesian reasoning, and introduces the {\alpha}-law as a principled diagnostic for monitoring update stability and reasoning quality in LLM inference systems.

Problem

• inference-time elicitation
• probability transformation
• large language models
• log-ratio relationship
• prompting pipelines

Innovation

• inference-time elicitation
• log-ratio transformation
• probability calibration
• large language models
• empirical characterization