The Interplay between Bayesian Inference and Conformal Prediction

Conformal prediction (CP) has emerged as a cutting-edge methodology in statistics and machine learning, providing prediction intervals with finite-sample frequentist coverage guarantees. Yet, its interplay with Bayesian statistics–often criticised for lacking frequentist guarantees–remains underexplored. Recent work has suggested that CP can “calibrate” Bayesian prediction regions, thereby imparting frequentist validity and motivating deeper investigation into frequentist-Bayesian hybrids. On the other side, Bayesian procedures have the potential to enhance CP with more informative intervals, toward nearly optimal solutions under a decision-theoretic framework. Thus, the two paradigms can be jointly used for a principled balance between validity and efficiency. This work provides a unified treatment of this emerging interface with open directions. After surveying existing ideas, we consolidate the literature with a Bayesian version of split CP, and present a simple analysis of the Binomial model, investigating prior’s role, efficiency, and computational complexity.