  Conditional selective inference requires an exact characterization of the
selection event, which is often unavailable except for a few examples like the
lasso. This work addresses this challenge by introducing a generic approach to
estimate the selection event, facilitating feasible inference conditioned on
the selection event. The method proceeds by repeatedly generating bootstrap
data and running the selection algorithm on the new datasets. Using the outputs
of the selection algorithm, we can estimate the selection probability as a
function of certain summary statistics. This leads to an estimate of the
distribution of the data conditioned on the selection event, which forms the
basis for conditional selective inference. We provide a theoretical guarantee
assuming both asymptotic normality of relevant statistics and accurate
estimation of the selection probability. The applicability of the proposed
method is demonstrated through a variety of problems that lack exact
characterizations of selection, where conditional selective inference was
previously infeasible.
