🤖 AI Summary
This work addresses the challenge of achieving conditional coverage in conformal prediction without relying on strong structural assumptions. To this end, it proposes the PIT-CP method, which post-processes arbitrary nonconformity scores via one-dimensional conditional density estimation—using tools such as mixture density networks or conditional normalizing flows—to map them into approximately feature-independent pivotal scores. The approach requires no stringent modeling assumptions while preserving marginal coverage and the geometric structure of prediction sets, and it substantially improves conditional coverage performance. Theoretical analysis provides both deterministic and high-probability upper bounds on the conditional coverage gap, along with formal guarantees on the volume and symmetric difference of the resulting prediction sets.
📝 Abstract
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.