  Conformal inference is a popular tool for constructing prediction intervals
(PI). We consider here the scenario of post-selection/selective conformal
inference, that is PIs are reported only for individuals selected from an
unlabeled test data. To account for multiplicity, we develop a general split
conformal framework to construct selective PIs with the false
coverage-statement rate (FCR) control. We first investigate the Benjamini and
Yekutieli (2005)'s FCR-adjusted method in the present setting, and show that it
is able to achieve FCR control but yields uniformly inflated PIs. We then
propose a novel solution to the problem, named as Selective COnditional
conformal Predictions (SCOP), which entails performing selection procedures on
both calibration set and test set and construct marginal conformal PIs on the
selected sets by the aid of conditional empirical distribution obtained by the
calibration set. Under a unified framework and exchangeable assumptions, we
show that the SCOP can exactly control the FCR. More importantly, we provide
non-asymptotic miscoverage bounds for a general class of selection procedures
beyond exchangeablity and discuss the conditions under which the SCOP is able
to control the FCR. As special cases, the SCOP with quantile-based selection or
conformal p-values-based multiple testing procedures enjoys valid coverage
guarantee under mild conditions. Numerical results confirm the effectiveness
and robustness of SCOP in FCR control and show that it achieves more narrowed
PIs over existing methods in many settings.
