Kernel-based optimally weighted conformal prediction intervals

Conformal prediction, originated in Vovk et al. [1999, 2005], offers a robust framework explicitly designed for reliable and distribution-free uncertainty quantification. Conformal prediction has become increasingly recognized and adopted within the domains of machine learning and statistics [Lei et al., 2013, Lei and Wasserman, 2014, Kim et al., 2020, Angelopoulos and Bates, 2023]. Assuming nothing beyond the exchangeability of data, conformal prediction excels in generating valid prediction sets under any given significance level, irrespective of the underlying data distribution and model assumptions. This capability makes it particularly valuable for uncertainty quantification in settings characterized by diverse and complex models. Going beyond the exchangeability assumption has been a research challenge, particularly as many real-world datasets (such as time-series data) are inherently non-exchangeable. Tibshirani et al. [2019] addresses situations where a feature distribution shifts between training and test data and restores valid coverage through weighted quantiles based on the likelihood ratio of the distributions. More recently, Barber et al. [2023] bounds the coverage gap using the total