Local asymptotics of selection models with applications in Bayesian selective inference
Contemporary focus on selective inference has renewed interest in the theory of selection models. In this paper, we analyze the asymptotic properties of selection models built on independent and identically distributed observations. We show that, under suitable regularity conditions, they behave asymptotically like a sequence of Gaussian selection models. This provides a natural generalization of the Local Asymptotic Normality framework of Le Cam (1960), and indicates a notion of local asymptotic selective normality as the appropriate simplifying theoretical framework for analysis of selective inference. As a key application, we consider the methodological consequences of the asymptotic theory for Bayesian selective inference. Specifically, we prove that the posterior distribution constructed from a selection model under a fixed prior is asymptotically equivalent to the posterior derived in the corresponding asymptotic Gaussian selection model under a uniform prior. Notably, the latter is often mis-calibrated in a frequentist sense, particularly for one-sided selection mechanisms. This demonstrates that the familiar asymptotic equivalence between Bayesian and frequentist approaches does not hold under selection.