While Large Language Models (LLMs) exhibit exceptional zero-shot relevance modeling, their high computational cost necessitates framing passage retrieval as a budget-constrained global optimization problem. Existing approaches passively rely on first-stage dense retrievers, which leads to two limitations: (1) failing to retrieve relevant passages in semantically distinct clusters, and (2) failing to propagate relevance signals to the broader corpus. To address these limitations, we propose Bayesian Active Learning with Gaussian Processes guided by LLM relevance scoring (BAGEL), a novel framework that propagates sparse LLM relevance signals across the embedding space to guide global exploration. BAGEL models the multimodal relevance distribution across the entire embedding space with a query-specific Gaussian Process (GP) based on LLM relevance scores. Subsequently, it iteratively selects passages for scoring by strategically balancing the exploitation of high-confidence regions with the exploration of uncertain areas. Extensive experiments across four benchmark datasets and two LLM backbones demonstrate that BAGEL effectively explores and captures complex relevance distributions and outperforms LLM reranking methods under the same LLM budget on all four datasets.

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