  In a general class of Bayesian nonparametric models, we prove that the
posterior distribution can be asymptotically approximated by a Gaussian
process. Our results apply to nonparametric exponential family that contains
both Gaussian and non-Gaussian regression, and also hold for both efficient
(root-n) and inefficient (non root-n) estimation. Our general approximation
theorem does not rely on posterior conjugacy, and can be verified in a class of
Gaussian process priors that has a smoothing spline interpretation [59, 44]. In
particular, the limiting posterior measure becomes prior-free under a Bayesian
version of "under-smoothing" condition. Finally, we apply our approximation
theorem to examine the asymptotic frequentist properties of Bayesian procedures
such as credible regions and credible intervals.
