  In supervised learning, including regression and classification, conformal
methods provide prediction sets for the outcome/label with finite sample
coverage for any machine learning predictor. We consider here the case where
such prediction sets come after a selection process. The selection process
requires that the selected prediction sets be `informative' in a well defined
sense. We consider both the classification and regression settings where the
analyst may consider as informative only the sample with prediction sets small
enough, excluding null values, or obeying other appropriate `monotone'
constraints. We develop a unified framework for building such informative
conformal prediction sets while controlling the false coverage rate (FCR) on
the selected sample. While conformal prediction sets after selection have been
the focus of much recent literature in the field, the new introduced
procedures, called InfoSP and InfoSCOP, are to our knowledge the first ones
providing FCR control for informative prediction sets. We show the usefulness
of our resulting procedures on real and simulated data.
