Model-free selective inference under covariate shift via weighted conformal <i>p</i> -values

Summary This paper introduces novel weighted conformal $ p $-values and methods for model-free selective inference. The problem is as follows: given test units with covariates $ X $ and missing responses $ Y $, how do we select units for which the responses $ Y $ are larger than user-specified values while controlling the proportion of false positives? Can we achieve this without modelling assumptions on the data and without restriction on the model for predicting the responses? Finally, methods should be applicable when there is a covariate shift between training and test data, which commonly occurs in practice. We answer these questions by first leveraging any prediction model to produce a class of well-calibrated weighted conformal $ p $-values, which control the Type-I error in detecting a large response. These $ p $-values cannot be passed onto classical multiple testing procedures since they may not obey a well-known positive dependence property. Hence, we introduce weighted conformal selection, a new procedure that controls the false discovery rate in finite samples. Besides prediction-assisted candidate selection, weighted conformal selection (i) allows us to infer multiple individual treatment effects, and (ii) extends to outlier detection with inlier distribution shifts. We demonstrate performance via simulations and applications to causal inference, drug discovery and outlier detection datasets.