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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 challenge and has yet to be fully solved. In this paper, we propose a novel flexible distribution-free method, PredSet-1Step, to construct prediction sets that can efficiently adapt to unknown covariate shift.
We formally show that our method is 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. This is a technical tool of independent interest.
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