Default Priors for the Smoothness Parameter in Gaussian Matérn Random Fields

Abstract

The Matérn family of covariance functions plays a prominent role in the analysis of geostatistical data due to its ability to model different smoothness behaviors. Although in many applications the smoothness parameter is set at an arbitrary value, a more satisfactory approach requires data–based inference about smoothness, especially in the light of new findings showing that the information this type of data has about the smoothness can be considerable in some settings. This work proposes a new class of easy–to–compute default priors for the parameters of a class of Gaussian random fields with Matérn covariance functions with unknown smoothness parameter. This class of priors is obtained by approximating a reference prior using the spectral representation of stationary random fields. This approximate reference prior has several advantages over the exact reference prior. First, the computation of the former is more stable and considerably less burdensome than that of the latter. Second, both the marginal prior of the smoothness parameter and the joint posterior of all parameters are proper for the approximate reference prior, while the status of these for the exact reference prior is currently unknown. Third, Bayesian inferences about the covariance parameters based on the approximate reference prior have satisfactory frequentist properties that are superior than those based on maximum likelihood. It was also found that a previously proposed ad–hoc prior for the smoothness and the approximate reference prior display in most cases similar statistical performance, with the former being computationally simpler. The methodology is illustrated with an analysis of rainfall totals in Switzerland.