Posterior distribution of nondifferentiable functions

Abstract: This paper examines the asymptotic behavior of the posterior distribution of a possibly nondifferentiable function ‘$g(\theta)$’, where $\theta$ is a finite-dimensional parameter of either a parametric or semiparametric model. The main assumption is that the distribution of a suitable estimator $\widehat{\theta}_n$, its bootstrap approximation, and the Bayesian posterior for $\theta$ all agree asymptotically.

It is shown that whenever $g$ is locally Lipschitz, though not necessarily differentiable, the posterior distribution of $g(\theta)$ and the bootstrap distribution of $g(\widehat{\theta}_n)$ coincide asymptotically. One implication is that Bayesians can interpret bootstrap inference for $g(\theta)$ as approximately valid posterior inference in a large sample. Another implication—built on known results about bootstrap inconsistency—is that credible intervals for a nondifferentiable parameter $g(\theta)$ cannot be presumed to be approximately valid confidence intervals (even when this relation holds true for $\theta$).