Objective Bayesian Analysis for a Spatial Model with Nugget Effects

Objective priors, especially reference priors, have been well-known for geostatistical data since Berger, et al. (2001). A long-standing problem is to develop objective Bayes inference for spatially correlated data with nugget effects. In this paper, objective priors for such cases are systematically studied. In addition to the Jeffreys priors and commonly used reference priors, two types of “exact” reference priors are derived based on improper marginal likelihoods, so 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. Surprisingly, the two type exact reference priors are identical. The properiety of the marginal priors and joint posteriors are studied for a large class of objective priors including all the objective priors developed. Interestingly, various Jeffreys and reference priors yield improper posteriors. Under general conditions, both Jeffreys-rule and “exact” reference prior result in proper posteriors. It is shown that the frequentist coverage probabilities of posterior credible intervals depend only on the shape paramater and the noice-to-signal ratio, but neither the regression coefficient nor the error variance. A simulation study is given to compare these two objective priors by frequentist coverage probabilities of the credible intervals. It is shown the exact reference priors performs much better then the Jeffreys-rule prior. Two real spatial datasets are used for illustration.