Random field priors for spectral density functions

In this paper we discuss how a Gaussian random field with Matérn covariance function can represent our prior uncertainty about the log-spectral density, $g(\omega)$, of a Gaussian, short memory time series. Hyperparameters can be suitably tuned in order to determine the mean square differentiability and the range of autocorrelation of the random field $g(\omega)$. However, Bayesian computations cannot be easily performed under such prior elicitations. We suggest therefore to approximate the Gaussian random field priors with a class of Gaussian Markov random fields which preserve the main features of the genuine prior distributions.