A Category-Theoretic Analysis of Conformal Prediction

TL;DR

This paper introduces a category-theoretic framework for conformal prediction, clarifying its structure, and connecting it with Bayesian and imprecise probabilistic methods, while also addressing privacy considerations.

Contribution

It develops a novel categorical approach to conformal prediction, decomposing its construction, and establishing links with Bayesian, frequentist, and privacy-preserving methods.

Findings

Decomposition of conformal regions into data-driven distribution sets and prediction regions.

Asymptotic convergence of conformal regions to Bayesian density level sets.

Conditions under which e-posteriors relate to conformal regions.

Abstract

Conformal prediction (CP) produces prediction regions with finite-sample, distribution free coverage guarantees, but its interpretation as a quantitative uncertainty tool is often left implicit. We develop a category-theoretic approach that makes this structure explicit. We show that Full Conformal Prediction can be represented as a morphism in two categories capturing (i) stability of set-valued procedures and (ii) measurability of random regions. Under mild conditions, we prove a commuting diagram result that decomposes the construction of a conformal region into two steps: Extracting a set of predictive distributions from the data, and then deriving a prediction region from this set. This decomposition provides a principled route to numerical uncertainty summaries beyond region size. We further prove an asymptotic compatibility result showing that, for Bayesian predictive scores in regular regimes, conformal regions converge to Bayesian predictive density level sets; We also provide quantitative rates under local empirical process and boundary regularity assumptions. This highlights a bridge between Bayesian, frequentist, and imprecise probabilistic prediction. We additionally identify conditions under which upper posterior constructions are related to e-posteriors, clarifying when e-value-based and conformal-imprecise representations can coincide. Finally, we show that the region extractor is functorial; This yields a modular privacy-compatible perspective in which privacy-preserving outer approximations of shared summary objects lead to conservative global prediction regions.