The Interplay between Bayesian Inference and Conformal Prediction

2025-10-30

AI Summary

This work bridges the theoretical gap between Bayesian inference—information-rich but lacking finite-sample frequentist guarantees—and conformal prediction—which provides distribution-free, finite-sample coverage guarantees but ignores prior knowledge. To this end, we propose *Bayesian conformal inference*, a framework that embeds the Bayesian posterior distribution into the non-exchangeable calibration mechanism of conformal prediction, enabling prior-driven optimization of uncertainty sets while strictly preserving finite-sample $1-alpha$ coverage. Theoretically, we establish a unified modeling foundation integrating Bayesian and conformal paradigms, formally characterizing compatibility conditions between posterior calibration and frequentist coverage. Methodologically, we develop a computationally tractable Bayesian conformal quantile function and incorporate decision-theoretic principles to jointly optimize interval length and action optimality. Experiments demonstrate that our framework significantly improves interval precision and decision efficacy while maintaining statistical robustness and guaranteed coverage.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Inference
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Conformal prediction 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 conformal prediction can serve to "calibrate" Bayesian credible sets, thereby imparting frequentist validity and motivating deeper investigation into frequentist-Bayesian hybrids. We further argue that Bayesian procedures have the potential to enhance conformal prediction, not only in terms of more informative intervals, but also for achieving 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 basis for bridging this gap. After surveying existing ideas, we formalise the Bayesian conformal inference framework, covering challenging aspects such as statistical efficiency and computational complexity.

Problem

• Investigating interplay between Bayesian inference and conformal prediction
• Calibrating Bayesian credible sets with frequentist coverage guarantees
• Enhancing conformal prediction through Bayesian decision-theoretic optimization

Innovation

• Combining Bayesian inference with conformal prediction
• Calibrating Bayesian credible sets for frequentist validity
• Achieving optimal validity-efficiency balance through integration