Conformal Prediction Sets Quantify Information Gain: A Theoretical Foundation (arXiv 2610.08785)

Forum topic · 小凯 · 2026-10-08

Summary

A new paper by Kevin Zhang and Stephen Bates (arXiv 2610.08785, October 2026) provides an information-theoretic foundation for using conformal prediction set size as a measure of uncertainty. Conformal prediction outputs prediction sets with finite-sample coverage guarantees, but the theoretical basis for interpreting set size as an uncertainty proxy was poorly understood. The authors introduce a family of generalized information measures, based on a decision-theoretic generalization of entropy for set-valued predictions, defined via conformal set size and coverage. Shannon mutual information admits an exact integral representation in terms of these measures. In standard classification settings, the reduction in conformal set size from additional information is bounded by calibration-dependent members of this family and satisfies a data processing inequality, up to finite-sample calibration and model error terms. The theory is validated on 11 classification settings, and in greedy feature selection, set-size reduction and Shannon mutual information can rank features differently.

Overview

Field: Machine Learning Authors: Kevin Zhang, Stephen Bates Published: 2026-10-06 arXiv: 2610.08785

Abstract (translated)

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. This work provides such a foundation using a decision-theoretic generalization of entropy tailored to set-valued prediction.

Specifically, the authors 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.

The paper shows that, in standard classification settings, the reduction in conformal set size from additional information:

1. Is sandwiched by calibration-dependent members of this family of measures; 2. Satisfies a data processing inequality;

in both cases up to finite-sample calibration and model error terms.

Significance

These results formally connect conformal prediction to classical information-theoretic quantities and justify using prediction set size reduction as a measure of information gain. The theory is validated empirically on 11 classification settings, where it is also shown that in greedy feature selection, set size reduction and Shannon mutual information can rank features differently.