This paper studies the problem of post-selection predictive inference in an online fashion. To avoid devoting resources to unimportant units, a preliminary selection of the current individual before reporting its prediction interval is common and meaningful in online predictive tasks. As a result of the temporal multiplicity introduced by the online selection process, it becomes essential to control the real-time false coverage-statement rate (FCR) which measures the overall miscoverage level. We develop a general framework named CAP (Calibration after Adaptive Pick), which performs an adaptive pick rule on historical data to construct a calibration set if the current individual is selected. This is followed by the output of a conformal prediction interval for the unobserved label. We present a series of tractable procedures for the construction of the calibration set for various popular online selection rules. It has been demonstrated that CAP achieves an exact selection-conditional coverage guarantee in the finite-sample and distribution-free regimes. In order to address the issue of the distribution shift in online data, we also embed CAP into some recent dynamic conformal prediction algorithms and prove that the proposed method can deliver long-run FCR control. Numerical results on both synthetic and real data corroborate that CAP can effectively control FCR around the target level and yield narrower prediction intervals than existing baselines across various settings.

Real-time decision-making gets more attention in the big data era. Here, we consider the problem of sample selection in the online setting, where one encounters a possibly infinite sequence of individuals collected over time with covariate information available. The goal is to select samples of interest that are characterized by their unobserved responses until the user-specified stopping time. We derive a new decision rule that enables us to find more preferable samples that meet practical requirements by simultaneously controlling two types of general constraints: individual and interactive constraints, which include the widely utilized False Selection Rate (FSR), cost limitations, and diversity of selected samples. The key elements of our approach involve quantifying the uncertainty of response predictions via predictive inference and addressing individual and interactive constraints in a sequential manner. Theoretical and numerical results demonstrate the effectiveness of the proposed method in controlling both individual and interactive constraints.

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.