How Conformal Prediction Sets Quantify Information Gain: An Information-Theoretic Foundation
符合性預測集合如何量化資訊增益：資訊理論的新視角
Conformal Prediction (CP) is widely used for uncertainty quantification because it guarantees finite-sample coverage, with prediction set size acting as an intuitive heuristic for uncertainty. This paper provides the missing information-theoretic foundation for this practice. The authors introduce a family of generalized information measures based on CP set size and coverage, showing that Shannon mutual information can be expressed exactly as an integral of these measures. Furthermore, they prove that the reduction in set size from additional information obeys a data processing inequality, theoretically justifying the use of CP set-size reduction as a metric for information gain.
Key points
Theoretical Foundation for Set Uncertainty
Provides the first rigorous decision-theoretic and information-theoretic proof justifying the common heuristic of using conformal set size as an uncertainty metric.
Exact Link to Shannon Mutual Information
Demonstrates that Shannon mutual information can be exactly represented as an integral over a family of generalized information measures derived from conformal sets.
Data Processing Inequality Obedience
Shows that the reduction in conformal prediction set size due to additional information obeys a data processing inequality, up to finite-sample calibration and model error.
Implications for Feature Selection
Empirical tests across 11 classifications show that set-size reduction and Shannon mutual information can rank features differently during greedy feature selection.
How it works
| 香農互資訊 (Shannon Mutual Information) | 符合性預測集合縮減 (Conformal Set-Size Reduction) | |
|---|---|---|
| Theoretical Foundation | 經典資訊理論與熵 (Classical Information Theory & Entropy) | 決策理論與符合性幾何 (Decision Theory & Conformal Geometry) |
| Guarantees | 漸進/總體期望值 (Asymptotic / Population Expected Values) | 有限樣本無分佈覆蓋保證 (Finite-Sample Distribution-Free Coverage) |
| Feature Evaluation Behavior | 衡量全域平均資訊量，可能偏好細粒度特徵 | 基於目標覆蓋率下的實質預測集合縮減，更具決策導向 |
Why it matters
Traditionally, conformal set size was treated merely as an intuitive heuristic for uncertainty. By mathematically linking it to Shannon information theory, this work elevates conformal prediction to a rigorous tool for measuring information gain. This enables researchers and practitioners to confidently use set-size reduction to quantify the 'value of information' in feature selection, active learning, and trustworthy AI, backed by formal mathematical guarantees.
Who it affects
- AI Researcher
- AI Developer
How to use it
- 1Using conformal set-size reduction to evaluate and rank the information gain of candidate features in feature selection tasks.
- 2Selecting the most informative data samples for labeling in active learning based on the expected reduction in conformal prediction set size.
Limitations & caveats
- Theoretical bounds and inequalities are subject to finite-sample calibration and model error terms.
- Empirical tests show that set-size reduction and Shannon mutual information can rank features differently, requiring careful choice based on the task objective.
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