Sharp training-conditional coverage for conformal prediction under covariate shift
Abstract
Weighted split conformal prediction reweights calibration scores by the likelihood ratio between the test and training covariate distributions and guarantees marginal coverage under covariate shift. We study its coverage conditional on the calibration data. An elementary argument, based on a single concentration inequality at a fixed population quantile, gives explicit training-conditional bounds without unspecified constants, and shows that the relevant scale is not the supremum of the likelihood ratio but a variance proxy built from the chi-squared divergence of the shift and from the average of the ratio over the part of the test population, of probability equal to the miscoverage level, where it is largest. A two-point lower bound shows that the root-m rate and the chi-squared contribution are intrinsic to the shift. Run at an explicitly inflated level, the weighted quantile becomes a deterministic PAC prediction set. We compare it with randomized rejection sampling and with importance-weighted learn-then-test and, through a certified choice of a clipping level for the likelihood ratio, map the regime in which each gives the narrower valid set. The analysis extends to estimated likelihood ratios and to tail functionals estimated from an unlabeled source sample, which yields a fully finite-sample certificate.
Related
- Conformal Prediction and Conditional Coverage for Tabular Foundation Models
- How Synthetic Labels Improve Conformal Prediction: A Perspective on Conditional Coverage
- Backward Conformal Prediction under Covariate Shift
- Finite-Sample Optimal Conformal Prediction Sets under Covariate Shift
- Sketched Calibration for Conformal Prediction under Covariate Shift
BibTeX
@misc{pournaderi2026sharp,
title = {{Sharp training-conditional coverage for conformal prediction under covariate shift}},
author = {Mehrdad Pournaderi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33456},
eprint = {2609.33456},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33456},
}
RIS
TY - UNPB
TI - Sharp training-conditional coverage for conformal prediction under covariate shift
AU - Mehrdad Pournaderi
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33456
UR - https://arxiv.org/abs/2609.33456
AB - Weighted split conformal prediction reweights calibration scores by the likelihood ratio between the test and training covariate distributions and guarantees marginal coverage under covariate shift. We study its coverage conditional on the calibration data. An elementary argument, based on a single concentration inequality at a fixed population quantile, gives explicit training-conditional bounds without unspecified constants, and shows that the relevant scale is not the supremum of the likelihood ratio but a variance proxy built from the chi-squared divergence of the shift and from the average of the ratio over the part of the test population, of probability equal to the miscoverage level, where it is largest. A two-point lower bound shows that the root-m rate and the chi-squared contribution are intrinsic to the shift. Run at an explicitly inflated level, the weighted quantile becomes a deterministic PAC prediction set. We compare it with randomized rejection sampling and with importance-weighted learn-then-test and, through a certified choice of a clipping level for the likelihood ratio, map the regime in which each gives the narrower valid set. The analysis extends to estimated likelihood ratios and to tail functionals estimated from an unlabeled source sample, which yields a fully finite-sample certificate.
ER -