Bayesian Post Training Enhancement of Regression Models with Calibrated Rankings

Accurate regression models are essential for scientific discovery, yet high-quality numeric labels are scarce and expensive. In contrast, rankings (especially pairwise) are easier to obtain from domain experts or artificial intelligence (AI) judges. We introduce Bayesian Enhancement with Calibrated Ranking (BAYES-ECR), a novel plug-and-play method that improves a base regressor’s prediction for a query by leveraging pairwise rankings between the query and reference items with known labels. BAYES-ECR performs a Bayesian update that combines a Gaussian likelihood from the regressor and the Bradley-Terry likelihood from the ranker. This yields a strictly log-concave posterior with a unique maximum likelihood solution and fast Newton updates. We show that prior state-of-the-art is a special case of our framework, and we identify a fundamental failure mode: Bradley-Terry likelihoods suffer from scale mismatch and curvature dominance when the number of reference items is large, which can degrade performance. From this analysis, we derive a calibration method to adjust the information originating from the expert rankings. BAYES-ECR shows a stunning 97.65% median improvement across 12 datasets over previous state-of-the-art method using a realistically-accurate ranker, and runs efficiently on a consumer-grade CPU.