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Conformal Prediction Regions are Imprecise Highest Density Regions

Michele Caprio, Yusuf Sale, Eyke Hüllermeier

TL;DR

The paper shows that the Imprecise Highest Density Region (IHDR) obtained from the credal set induced by a consonant conformal transducer $\pi$ coincides with the classical Conformal Prediction Region $\mathscr{R}_\alpha(\mathbf{y}^n)$ and preserves the $1-\alpha$ coverage uniformly across exchangeable data. By linking CP to Imprecise Probability (IP) via upper probabilities $\overline{\Pi}$ and clouds, it reveals that consonant plausibility functions are monoid homomorphisms and that IHDRs provide a model-free IP interpretation of CP guarantees. The main result establishes a precise equivalence between IHDRs and CPRs, offering a unified view of CP within IP theory and highlighting algebraic properties of consonance. The work also analyzes how changing the nonconformity measure $\Psi$ affects region size and discusses the geometric and practical implications of driving the credal set toward boundary points under consonance, pointing to directions for future research in CP geometry and consonance limitations.

Abstract

Recently, Cella and Martin proved how, under an assumption called consonance, a credal set (i.e. a closed and convex set of probabilities) can be derived from the conformal transducer associated with transductive conformal prediction. We show that the Imprecise Highest Density Region (IHDR) associated with such a credal set corresponds to the classical Conformal Prediction Region. In proving this result, we establish a new relationship between Conformal Prediction and Imprecise Probability (IP) theories, via the IP concept of a cloud. A byproduct of our presentation is the discovery that consonant plausibility functions are monoid homomorphisms, a new algebraic property of an IP tool.

Conformal Prediction Regions are Imprecise Highest Density Regions

TL;DR

The paper shows that the Imprecise Highest Density Region (IHDR) obtained from the credal set induced by a consonant conformal transducer coincides with the classical Conformal Prediction Region and preserves the coverage uniformly across exchangeable data. By linking CP to Imprecise Probability (IP) via upper probabilities and clouds, it reveals that consonant plausibility functions are monoid homomorphisms and that IHDRs provide a model-free IP interpretation of CP guarantees. The main result establishes a precise equivalence between IHDRs and CPRs, offering a unified view of CP within IP theory and highlighting algebraic properties of consonance. The work also analyzes how changing the nonconformity measure affects region size and discusses the geometric and practical implications of driving the credal set toward boundary points under consonance, pointing to directions for future research in CP geometry and consonance limitations.

Abstract

Recently, Cella and Martin proved how, under an assumption called consonance, a credal set (i.e. a closed and convex set of probabilities) can be derived from the conformal transducer associated with transductive conformal prediction. We show that the Imprecise Highest Density Region (IHDR) associated with such a credal set corresponds to the classical Conformal Prediction Region. In proving this result, we establish a new relationship between Conformal Prediction and Imprecise Probability (IP) theories, via the IP concept of a cloud. A byproduct of our presentation is the discovery that consonant plausibility functions are monoid homomorphisms, a new algebraic property of an IP tool.

Paper Structure

This paper contains 10 sections, 6 theorems, 28 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conformal Prediction
  4. Imprecise Probabilities
  5. Relationship between Belief Functions and TCP
  6. Imprecise Highest Density Regions
  7. Building the Equivalent Prediction Region
  8. Varying the Nonconformity Measure $\Psi$
  9. Discussion
  10. Conclusion

Key Result

Lemma 1

A consonant plausibility function $pl$ is a monoid homomorphism between the monoids $(\Sigma_{\mathbb{Y}}, \cup)$ and $([0, 1], \oplus)$, where $\cup$ is the set union operation and $\oplus$ is the tropical addition on $[0, 1]$

Figures (2)

  • Figure 1: Top: Our proposed, "indirect" methodology to derive a prediction region. We first use the consonant conformal transducer $\pi$ to derive credal set $\mathcal{M}(\overline{\Pi})$, and then extract from the latter the IHDR $\text{IR}_\alpha$. Bottom: Classical CP methodology, in which the Conformal Prediction Region is obtained as in \ref{['eq_imp5']}.
  • Figure 2: Visual representation of $\mathcal{M}(\overline{\Pi}_\mathbf{y^n})$ in our example. As we can see, it is "pushed" towards the boundary of the unit simplex. We also depicted $p^\text{emp}=(0.2,0.3,0.5)^\top$ as a black dot.

Theorems & Definitions (14)

  • Lemma 1
  • proof
  • Lemma 2: Properties of a Consonant Upper Probability
  • proof
  • Remark 3
  • Definition 4: Imprecise Highest Density Region coolen1992imprecise
  • Proposition 5: IHDR Corresponds to the Conformal Prediction Region
  • proof
  • Proposition 6: Elements of the IHDR
  • proof
  • ...and 4 more