A Unified Theory of Conditional Coverage in Conformal Prediction with Applications

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.