Modern retrieval systems combine heterogeneous relevance signals — lexical scores such as BM25, dense vector similarities, and learned re-rankers — yet these signals live on incompatible scales and carry no shared probabilistic meaning. A BM25 score of 8.4 and a cosine similarity of 0.85 cannot be added, averaged, or thresholded together without resorting to ad-hoc normalization or rank-based fusion that discards magnitude information. This paper develops a single Bayesian principle that resolves the problem for every signal at once: each raw score is the observable of a binary relevance hypothesis, and the calibrated probability of relevance is the posterior obtained from the score's log-likelihood ratio plus an independent prior, evaluated in log-odds space. We instantiate this principle twice. For lexical retrieval, a parametric sigmoid likelihood and a corpus-level base rate make the posterior additive in log-odds and reduce expected calibration error by 68–77% without relevance labels. For dense retrieval, a nonparametric likelihood ratio between a local relevant-distance density and a global background density calibrates vector scores using only the distributional statistics that an approximate-nearest-neighbor index already computes; a cross-modal conditional-independence argument breaks the circularity inherent in estimating the relevant density. Because each signal's evidence is a log-likelihood ratio on a common Bernoulli log-odds scale, fusion is additive: the evidence terms sum, with the prior added once. We develop a log-odds conjunction that aggregates calibrated evidence while avoiding the shrinkage pathology of naive probabilistic conjunction — exactly a normalized Logarithmic Opinion Pool — together with a query-adaptive weighting that lets the per-signal reliabilities depend on query features. The result is a single additive posterior in log-odds that fuses an arbitrary number of conditionally independent signals — here, sparse and dense — with no re-calibration of the existing signals, and preserves the safe dynamic pruning (WAND, Block-Max WAND) of the underlying BM25 retrieval. On five BEIR datasets the framework is competitive with, and slightly exceeds on aggregate, the strongest tuning-free baselines (convex combination, reciprocal rank fusion) while additionally producing calibrated probabilities (best zero-shot NDCG@10 +6.28 over BM25; the base-rate correction reduces expected calibration error by 68–77% without relevance labels).
Jaepil Jeong (Fri,) studied this question.