Optimal Bayesian Stopping for Efficient Inference of Consistent LLM Answers

A simple strategy for improving LLM accuracy, especially in math and reasoning problems, is to sample multiple responses and submit the answer most consistently reached. In this paper we leverage Bayesian prior information to save on sampling costs, stopping once sufficient consistency is reached. Although the exact posterior is computationally intractable, we further introduce an efficient ``$L$-aggregated'' stopping policy that tracks only the $L-1$ most frequent answer counts. Theoretically, we prove that $L=3$ is all you need: this coarse approximation is sufficient to achieve asymptotic optimality, and strictly dominates prior-free baselines, while having a fast posterior computation. Empirically, this identifies the most consistent (i.e., mode) LLM answer and achieves similar answer accuracy using fewer samples.