Pseudo-marginal Sampling for Selective Inference

arXiv:2611.13492v1 Announce Type: cross Abstract: Inference after adaptive selection must account for selection bias. One approach is to condition on the selected event, but for complex selection procedures, the resulting conditional distribution can be analytically intractable and require sampling-based methods. Rejection sampling provides a natural baseline: it repeatedly draws from a proposal and retains only samples that reproduce the observed selection. This can be prohibitively expensive when the selection event is rare. We address this computational bottleneck with a pseudo-marginal Markov chain Monte Carlo algorithm that samples from the selective distribution. Using a weak Poincar\'e inequality, we analyze its computational efficiency relative to rejection sampling and identify when it can offer substantial gains. In the classical file-drawer problem, we derive precise computational costs for both methods, illustrating exponential improvement as the selection event becomes increasingly rare. In numerical experiments involving complex selection procedures without tractable conditional distributions, the pseudo-marginal sampler is often hundreds of times more efficient than rejection sampling, while the resulting tests achieve nominal selective Type I error.