  In this work, we provide a refinement of the selective CLT result of Tian and
Taylor (2015), which allows for selective inference in non-parametric settings
by adjusting for the asymptotic Gaussian limit for selection. Under some
regularity assumptions on the density of the randomization, including heavier
tails than Gaussian satisfied by e.g. logistic distribution, we prove the
selective CLT holds without any assumptions on the underlying parameter,
allowing for rare selection events. We also show a selective CLT result for
Gaussian randomization, though the quantitative results are qualitatively
different for the Gaussian randomization as compared to the heavier tailed
results. Furthermore, we propose a bootstrap version of this test statistic,
which is provably asymptotically pivotal uniformly across a family of
non-parametric distributions. This result can be interpreted as resolving the
impossibility results of Leeb and Potscher (2006). We describe several sampling
methods involving the projected Langevin Monte Carlo to compute the
bootstrapped test statistic and the corresponding confidence intervals valid
after selection. The applications of our work include valid inferential and
sampling tools after running various model selection algorithms including their
combinations into multiple views/queries framework. We also present a way to do
data carving, providing more powerful tests than classical data splitting by
reusing the information in the data from the first stage.
