Predicting sets of outcomes -- instead of unique outcomes -- is a promising solution to uncertainty quantification in statistical learning. Despite a rich literature on constructing prediction sets with statistical guarantees, adapting to unknown covariate shift -- a prevalent issue in practice -- poses a serious unsolved challenge. In this paper, we show that prediction sets with finite-sample coverage guarantee are uninformative and propose a novel flexible distribution-free method, PredSet-1Step, to efficiently construct prediction sets with an asymptotic coverage guarantee under unknown covariate shift. We formally show that our method is \textit{asymptotically probably approximately correct}, having well-calibrated coverage error with high confidence for large samples. We illustrate that it achieves nominal coverage in a number of experiments and a data set concerning HIV risk prediction in a South African cohort study. Our theory hinges on a new bound for the convergence rate of the coverage of Wald confidence intervals based on general asymptotically linear estimators.

翻译：预测结果集合——而非单一结果——是统计学习中不确定性量化的一个有前景的解决方案。尽管已有大量文献致力于构建具有统计保证的预测集，但适应未知协变量漂移（实际中普遍存在的问题）仍然是一个尚未解决的严峻挑战。本文证明，具有有限样本覆盖保证的预测集在实际中无信息性，并提出一种新颖的灵活无分布方法——PredSet-1Step，以高效构建在未知协变量漂移下具有渐近覆盖保证的预测集。我们严格证明了该方法具有\textit{渐近概率近似正确}性质，即对大样本能以高置信度实现良好校准的覆盖误差。通过多项实验及一项关于南非队列研究中HIV风险预测的数据集，我们验证了该方法能达到名义覆盖水平。我们的理论依赖于基于一般渐近线性估计量的Wald置信区间覆盖率收敛速度的新界。