In the Retrieval-Augmented Generation (RAG) system, advanced Large Language Models (LLMs) have emerged as effective Query Likelihood Models (QLMs) in an unsupervised way, which re-rank documents based on the probability of generating the query given the content of a document. However, directly prompting LLMs to approximate QLMs inherently is biased, where the estimated distribution might diverge from the actual document-specific distribution. In this study, we introduce a novel framework, $\mathrm{UR^3}$, which leverages Bayesian decision theory to both quantify and mitigate this estimation bias. Specifically, $\mathrm{UR^3}$ reformulates the problem as maximizing the probability of document generation, thereby harmonizing the optimization of query and document generation probabilities under a unified risk minimization objective. Our empirical results indicate that $\mathrm{UR^3}$ significantly enhances re-ranking, particularly in improving the Top-1 accuracy. It benefits the QA tasks by achieving higher accuracy with fewer input documents.

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