Bayesian Conformal Prediction as a Decision Risk Problem

We propose an optimised Bayesian Conformal Prediction (BCP) framework that selects the conformal threshold via a decision-theoretic risk minimisation criterion. BCP uses Bayesian posterior predictive densities as non-conformity scores and Bayesian quadrature to estimate and minimise the expected prediction set size. Operating within a split conformal framework, BCP provides valid coverage guarantees and demonstrates reliable empirical coverage under model misspecification. Across regression and classification tasks, including distribution-shifted settings such as ImageNet-A, BCP yields prediction sets of comparable size to split conformal prediction, while exhibiting substantially lower run-to-run variability in set size. In sparse regression with nominal coverage of 80 percent, BCP achieves 81 percent empirical coverage under a misspecified prior, whereas Bayesian credible intervals under-cover at 49 percent.