Objective Bayesian Analysis of Geometrically Anisotropic Spatial Data

Anisotropic models are often used in spatial statistics to analyze spatially referenced data. Within a Bayesian framework we develop default priors for the anisotropic Gaussian random field model with and without including a nugget parameter accounting for the effects of microscale variations and measurement errors. We present Jeffreys priors and a reference prior and study their posterior propriety. Moreover, we obtain that the predictive distributions at ungauged locations have finite variance. We also show that the seemingly uninformative uniform prior for the anisotropy parameters, ratio and angle, yields an improper posterior. Finally, we find that the proposed priors have good frequentist properties and we illustrate our approach by analyzing two data sets for which we discuss model choice as well as predictions and uncertainty estimates.