In this master thesis we propose a nonparametric Bayesian approach to estimating dependency in stationary time series and in second-order stationary spatial models over continuous domains.Our main focus will be the estimation of the unknown covariance function in these models. We will apply nonparametric Bayesian methods to the problem by working with the corresponding spectral measures within the frequency domain, the reason for this is that this is an easy way to always be sure that we are only working with well-behaved and valid covariance functions (where valid means positive definite).
The thesis is divided into two main parts in which we treat covariance functions for stationary Gaussian time series and isotropic covariance functions for second-order stationary Gaussian random fields separately. In both sections we will derive reasonable models, explain how we can obtain and construct flexible and subjective prior distributions and we will show that it is possible and also fairly easy to obtain posterior inference based on simulation routines. For the stationary Gaussian time series models we are also able to derive the asymptotic posterior properties for the unknown spectral measure and covariance function, by the use of the Whittle approximation and other related approximations to the full multivariate Gaussian likelihood. Some of these results become mirror results, in the Bernstein-von Mises sense, of the classical frequentist solutions.