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

TL;DR

This paper uncovers a multiplicative scaling law governing how large language models revise their beliefs, providing a theoretical framework and empirical evidence for the stability and dynamics of belief updates during inference.

Contribution

It introduces the { extalpha}-law, a principled scaling law for observable belief revision in LLMs, linking theoretical stability conditions with empirical behavior across multiple models and benchmarks.

Findings

Models exhibit near-Bayesian update behavior in single steps.

Multi-step revisions show decreasing exponent, indicating stable long-term dynamics.

Architecture-specific patterns influence how models weigh prior beliefs and new evidence.

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.