Bernstein von Mises Theorems in Wasserstein distance

We study the Bernstein von-Mises (BvM) phenomenon in Gaussian process regression models by retaining the leading terms of the induced Karhunen--Loeve expansion. A recent related result by Bontemps, 2011 in a sieve prior context necessitates the prior to be flat, ruling out commonly used Gaussian process models where the prior flatness is determined by the decay rate of the eigenvalues of the covariance kernel. We establish the BvM phenomena in the L_2 Wasserstein distance instead of the commonly used total variation distance, thereby encompassing a wide class of practically useful Gaussian process priors. We also develop a general technique to derive posterior rates of convergence from Wasserstein BvMs and apply it to Gaussian process priors with unbounded covariate domain. Specific illustrations are provided for the squared-exponential covariance kernel.