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 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 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 sets for a nondifferentiable parameter g(theta) cannot be presumed to be approximately valid confidence sets (even when this relation holds true for theta).
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