Objective Bayesian analysis for a spatial model with nugget effects

The focus of this paper is objective priors for spatially correlated data with nugget effects. In addition to the Jeffreys priors and commonly used reference priors, two types of “exact” reference priors are derived based on improper marginal likelihoods. An “equivalence” theorem is developed in the sense that the expectation of any function of the score functions of the marginal likelihood function can be taken under marginal likelihoods. Interestingly, these two types of reference priors are identical.The propriety of the marginal priors and joint posteriors is studied for a large class of objective priors including all objective priors developed. Under quite general conditions, only Jeffreys-rule and “exact” reference priors yield proper posteriors. A simulation study shows that the exact reference priors perform better than the Jeffreys-rule prior. Two real spatial datasets are used for illustration.