While Conformal Prediction (CP) has proven to be a powerful framework for uncertainty quantification, guaranteeing conditional coverage remains a central 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, interpretability, 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 highly accurate point estimates. 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 bounds on the conditional coverage gap, alongside volumetric and symmetric-difference bounds. We present known minimax-optimal conditional estimation techniques while also motivating the use of modern conditional density estimators, including Mixture Density Networks and Conditional Normalizing Flows. Finally, we empirically demonstrate on various datasets that our PIT-CP procedure matches or outperforms many state-of-the-art conformal prediction strategies with minimal effort and computational cost.

翻译：尽管共形预测已被证明是一种强大的不确定性量化框架，但保证条件覆盖仍然是一个核心挑战。尽管已知在无结构假设的情况下，有限样本、分布自由的条件有效性不可能实现，但我们证明其本质上等价于构造一个得分分布与特征无关的非一致分数。这一理论刻画催生了PIT-CP——一种新的后处理校正方法，可将任意基非一致分数映射为近似不变的分数，同时保留其几何结构、可解释性和边际覆盖。该视角在实践中尤为吸引人：当已有预测驱动模型能提供高度精确的点估计时，重新训练完整的生成模型既不经济也不高效。我们的方法将问题简化为对诱导分数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这一变换，推导了条件覆盖差距的界，同时给出了体积和对称差界。我们介绍了已知的极小极大最优条件估计技术，同时倡导使用现代条件密度估计器，包括混合密度网络和条件归一化流。最后，我们在多个数据集上通过实验证明，PIT-CP方法能以最小代价和计算成本匹配或超越多种最先进的共形预测策略。