$\texttt{BetaConform}$: Efficient MAP Estimation of LLM Ensemble Judgment Performance with Prior Transfer

LLM ensembles are widely used for LLM judges. However, how to estimate their accuracy, especially in an efficient way, is unknown. In this paper, we present a principled $\textit{maximum a posteriori}$ (MAP) framework for an economical and precise estimation of the performance of LLM ensemble judgment. We first propose a mixture of Beta-Binomial distributions to model the judgment distribution, revising from the vanilla Binomial distribution. Next, we introduce a conformal prediction-driven approach that enables adaptive stopping during iterative sampling to balance accuracy with efficiency. Furthermore, we design a prior transfer mechanism that utilizes learned distributions on open-source datasets to improve estimation on a target dataset when only scarce annotations are available. Finally, we present $\texttt{BetaConform}$, a framework that integrates our distribution assumption, adaptive stopping, and the prior transfer mechanism to deliver a theoretically guaranteed distribution estimation of LLM ensemble judgment with minimum labeled samples. $\texttt{BetaConform}$ is also validated empirically. For instance, with only $10$ samples from the TruthfulQA dataset, for a Llama ensembled judge, $\texttt{BetaConform}$ gauges its performance with an error margin as small as $3.37\\%$.