Posterior distribution of nondifferentiable functions

This paper examines the asymptotic behavior of the posterior distribution of a possibly nondifferentiable function g(theta), where theta is a finite dimensional parameter. The main assumption is that the distribution of the maximum likelihood estimator theta_n, its bootstrap approximation, and the Bayesian posterior for theta all agree asymptotically. It is shown that whenever g is Lipschitz, though not necessarily differentiable, the posterior distribution of g(theta) and the bootstrap distribution of g(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 the posterior distribution of g(theta) does not coincide with the asymptotic distribution of g(theta_n) at points of nondifferentiability. Consequently, frequentists cannot presume that credible sets for a nondifferentiable parameter g(theta) can be interpreted as approximately valid confidence sets (even when this relation holds true for theta).