While Conformal Prediction (CP) has proven to be a powerful framework for uncertainty quantification, guaranteeing conditional coverage remains a fundamental challenge. Although finite-sample, distribution-free conditional validity is known to be impossible without structural assumptions, we show that it is fundamentally equivalent to constructing a nonconformity score whose distribution is independent of the features. This theoretical characterization motivates PIT-CP, a new post-processing correction that maps any base nonconformity score to an approximately invariant one while preserving its geometry, potential interpretability properties, and marginal coverage. This perspective is particularly appealing in practice, since it may be neither economical nor time-effective to retrain a full generative model when a strong prediction-driven model already provides a highly accurate point estimate. Our procedure reduces the problem to one-dimensional conditional density estimation on the induced score, rather than full conditional density estimation on the original outcome space. We show how to estimate this transform in practice and derive upper bounds on the conditional coverage gap, both deterministically and with high probability. We also establish volumetric and symmetric-difference bounds. Finally, we state known minimax-optimal conditional estimation distance bounds, while also motivating the use of modern state-of-the-art conditional density estimators, including Mixture Density Networks and Conditional Normalizing Flows.

翻译：尽管共形预测（CP）已被证明是一种强大的不确定性量化框架，但保证条件覆盖仍是一个基础性挑战。虽然已知在缺乏结构假设的情况下，无法实现有限样本下的分布自由条件有效性，但我们证明其本质上等价于构造一个分布独立于特征的奇异度评分。这一理论刻画催生了PIT-CP——一种新的后处理校正方法，它将任意基础奇异度评分映射为近似不变评分，同时保留其几何结构、潜在可解释性及边缘覆盖。该视角在实践中极具吸引力，因为当强预测驱动模型已提供高度精确的点估计时，重新训练完整生成模型既不经济也不高效。我们的方法将问题降维至对诱导评分进行一维条件密度估计，而非在原始结果空间上进行完整条件密度估计。我们展示了如何在实践中估计该变换，并推导了条件覆盖间隙的上界（包括确定性上界和高概率上界），同时建立了体积界和对称差异界。最后，我们列举了已知的极小化最优条件估计距离界，并论证了使用现代最先进条件密度估计器（包括混合密度网络和条件归一化流）的合理性。