Conformal Prediction Sets Quantify Information Gain: A Theoretical Perspective

2026-10-06

AI Summary

This study addresses the lack of an information-theoretic foundation for using conformal prediction set size as a measure of uncertainty. Drawing on decision-theoretic generalized entropy, this work introduces a family of generalized information measures that integrate set size with coverage probability, yielding an exact integral representation of Shannon mutual information. It thereby establishes a rigorous theoretical connection between conformal prediction sets and information gain, proving that the proposed measure satisfies the data processing inequality. The theoretical findings are validated across eleven classification tasks, revealing discrepancies in feature ranking between the two classes of metrics. Ultimately, this research provides formal theoretical justification for employing conformal prediction set size reduction as a legitimate proxy for information gain.

Abstract

Conformal prediction is a popular tool for uncertainty quantification that outputs prediction sets with finite-sample coverage guarantees. While prediction set size is commonly used as a heuristic measure of uncertainty, the information-theoretic basis for this interpretation remains poorly understood. In this work, we provide such a foundation using a decision-theoretic generalization of entropy tailored to set-valued prediction. In particular, we introduce a family of generalized information measures based on the size and coverage of conformal prediction sets. Notably, Shannon mutual information admits an exact integral representation in terms of these measures. We then show that, in standard classification settings, the reduction in conformal set size from additional information (i) is sandwiched between calibration-dependent members of this family and (ii) obeys a data processing inequality, both up to finite-sample calibration and model error terms. Together, our results formally relate conformal prediction to classical information-theoretic quantities and justify using set-size reduction as an information gain metric. Empirically, we validate our theory across 11 classification settings and show that set-size reduction and Shannon mutual information can rank features differently in a greedy feature selection experiment.

Problem

• Conformal Prediction
• Uncertainty Quantification
• Information Gain
• Shannon Mutual Information
• Prediction Set Size

Innovation

• Conformal Prediction
• Information Gain
• Shannon Mutual Information
• Generalized Entropy
• Data Processing Inequality