  Inspired by sample splitting and the reusable holdout introduced in the field
of differential privacy, we consider selective inference with a randomized
response. We discuss two major advantages of using a randomized response for
model selection. First, the selectively valid tests are more powerful after
randomized selection. Second, it allows consistent estimation and weak
convergence of selective inference procedures. Under independent sampling, we
prove a selective (or privatized) central limit theorem that transfers
procedures valid under asymptotic normality without selection to their
corresponding selective counterparts. This allows selective inference in
nonparametric settings. Finally, we propose a framework of inference after
combining multiple randomized selection procedures. We focus on the classical
asymptotic setting, leaving the interesting high-dimensional asymptotic
questions for future work.
