Properties of the posterior distribution of a regression model based on Gaussian random fields

We consider the regression problem based on Gaussian processes. We assume that the prior distribution on the vector of parameters in the corresponding model of the covariance function is non-informative. Under this assumption, we prove the Bernstein–von Mises theorem that states that the posterior distribution on the parameters vector is close to the corresponding normal distribution. We show results of numerical experiments that indicate that our results apply in practically important cases.