Distributional Balancing Weights for Conformal Calibration under Covariate Shift

Conformal prediction provides distribution-free marginal coverage under exchangeability. Under covariate shift, covariate distributions change while the conditional response distribution remains unchanged. Weighted conformal prediction corrects this mismatch using known density ratios, which can be difficult to estimate. We propose conformal calibration with distributional balancing weights, learned to align source calibration and unlabeled target covariate distributions without using calibration or target responses. Characteristic function distance provides a common framework for kernel-based and energy-based balancing. We establish a finite-sample error bound for estimating the target score distribution, yielding approximate marginal coverage guarantees. The bound separates covariate imbalance, approximation of conditional score probabilities, target-sample uncertainty, and weight concentration. We connect this bound to weight optimization and develop a validation rule for selecting weights without responses. A spectral analysis shows that nearly uniform fitted weights can miss increasingly fine-scale shifts and yield persistent undercoverage despite bounded density ratios. Controlled synthetic and semi-synthetic experiments reproduce this failure and show that tuning bandwidth and regularization can restore near-nominal coverage. Together, these results highlight the importance of kernel choice and regularization beyond controlling weight concentration alone.