  We study 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. Since the online selection
causes a temporal multiplicity in the selected prediction intervals, it is
important 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) that performs an adaptive pick rule on
historical data to construct a calibration set if the current individual is
selected and then outputs a conformal prediction interval for the unobserved
label. We provide tractable procedures for constructing the calibration set for
popular online selection rules. We proved that CAP can achieve an exact
selection-conditional coverage guarantee in the finite-sample and
distribution-free regimes. To account for the distribution shift in online
data, we also embed CAP into some recent dynamic conformal prediction
algorithms and show 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 more narrowed
prediction intervals over existing baselines across various settings.
