Conformalized Multiple Testing after Data-dependent Selection

The task of distinguishing individuals of interest from a vast pool of candidates using predictive models has garnered significant attention in recent years. This task can be framed as a conformalized multiple testing procedure, which aims at quantifying prediction uncertainty by controlling the false discovery rate (FDR) via conformal inference. In this paper, we tackle the challenge of conformalized multiple testing after data-dependent selection procedures. To guarantee the construction of valid test statistics that accurately capture the distorted distribution resulting from the selection process, we leverage a holdout labeled set to closely emulate the selective distribution. Our approach involves adaptively picking labeled data to create a calibration set based on the stability of the selection rule. This strategy ensures that the calibration data and the selected test unit are exchangeable, allowing us to develop valid conformal p-values. Implementing with the famous Benjamini-Hochberg (BH) procedure, it effectively controls the FDR over the selected subset. To handle the randomness of the selected subset and the dependence among the constructed p-values, we establish a unified theoretical framework. This framework extends the application of conformalized multiple testing to complex selective settings. Furthermore, we conduct numerical studies to showcase the effectiveness and validity of our procedures across various scenarios.