Posterior contraction in Gaussian process regression using Wasserstein approximations

We study posterior rates of contraction in Gaussian process regression with potentially unbounded covariate domain. Our argument relies on developing a Gaussian approximation to the posterior of the leading coefficients of a Karhunen–Loève expansion of the Gaussian process. The salient feature of our result is deriving such an approximation in the L2 Wasserstein distance and relating the speed of the approximation to the posterior contraction rate using a coupling argument. Specific illustrations are provided for the Gaussian (or squared exponential) and Matérn covariance kernels.