Objective Bayesian analysis of spatially correlated data

Spatially varying phenomena are often modeled using Gaussian random fields, specified by their mean function and covariance function. The spatial correlation structure of these models is commonly specified to be of a certain form (e.g., spherical, power exponential, rational quadratic, or Matern) with a small number of unknown parameters. We consider objective Bayesian analysis of such spatial models, when the mean function of the Gaussian random field is specified as in a linear model. It is thus necessary to determine an objective (or default) prior distribution for the unknown mean and covariance parameters of the random field. We first show that common choices of default prior distributions, such as the constant prior and the independent Jeffreys prior, typically result in improper posterior distributions for this model. Next, the reference prior for the model is developed and is shown to yield a proper posterior distribution. A further attractive property of the reference prior is that it can be used directly for computation of Bayes factors or posterior probabilities of hypotheses to compare different correlation functions, even though the reference prior is improper. An illustration is given using a spatial dataset of topographic elevations.