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.

翻译：共形预测提供了有限样本的边缘有效性，但许多应用要求覆盖能够适应异质性测试点或子群体。现有针对条件覆盖的方法大多采用逐例分析，缺乏关于渐近条件有效性如何产生、不同方法应如何比较、以及此类保证如何扩展到结构化数据的一般性理论。我们建立了一个针对条件覆盖的共形方法统一框架与理论。在此框架内，我们通过两种互补路径推导出条件误覆盖的非渐近界：一种用于直接得分控制的逐点路径，以及一种用于分位数中心化方法的$L_p$路径。该理论阐明了控制渐近条件有效性的误差来源，提供了对现有方法的统一解释，并支持面向条件覆盖的模型选择、协变量偏移下的局部化、以及基于加权对称性的结构化数据设置等应用与扩展。数值结果支持理论结论。