A Unified Theory of Conditional Coverage in Conformal Prediction with Applications

TL;DR

This paper develops a unified theoretical framework for conformal prediction methods that aim to provide conditional coverage guarantees, addressing heterogeneity in test points and structured data.

Contribution

It introduces a comprehensive theory for asymptotic conditional validity, compares different procedures, and extends conformal methods to structured data and model selection.

Findings

Derived non-asymptotic bounds for conditional miscoverage.

Unified interpretation of existing conformal methods.

Numerical results support theoretical insights.

Abstract

Conformal prediction provides finite-sample marginal validity, but many applications require coverage that adapts to heterogeneous test points or subpopulations. Existing methods for conditional coverage are largely analyzed case by case, leaving limited general theory for how asymptotic conditional validity arises, how different procedures should be compared, and how such guarantees extend to structured data. We develop a unified framework and theory for conformal methods targeting conditional coverage. Within this framework, we derive non-asymptotic bounds for conditional miscoverage through two complementary routes: a pointwise route for direct score control and an $L_p$ route for quantile-centered methods. The theory clarifies the error sources governing asymptotic conditional validity, yields a common interpretation of existing methods, and supports applications and extensions to conditional-coverage-oriented model selection, localization under covariate shift, structured-data settings through a weighted symmetry-based formulation and more. Numerical results support the theoretical conclusions.