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Abstract: The conditional autoregressive (CAR) and simultaneous autoregressive (SAR) models both have been used extensively for the analysis of spatial structure underlying lattice data in many areas, such as epidemiology, demographics, economics, and geography. Default Bayesian analyses have been conducted recently, but the Bayesian approach has not used or explored these two models with nugget effects. In this paper, we consider general autoregressive models including both CAR and SAR models. The Jeffreys-rule, independence Jeffreys, commonly used reference and “exact” reference priors are derived. The propriety of the marginal priors and joint posteriors is studied for a large class of objective priors. Various Jeffreys and reference priors are shown to yield improper posteriors and only the Jeffreys-rule and the “exact” reference priors yield proper posteriors. We make comparisons for these two objective priors using the frequentist coverage probabilities of the credible intervals. An illustration is given using a real spatial data-set. [Copyright &y& Elsevier]

Abstract: The simultaneously autoregressive model (abbreviated as SAR) has been extensively applied for lattice (regional summary) data. A Bayesian approach has been studied by De Oliveira and Song (2008), but they only considered two versions of Jeffreys priors, Jeffreys-rule and independence Jeffreys priors. They recommended the independence Jeffreys prior for a default prior. This prior is known to have the potential problem of posterior impropriety. In this paper, we consider the reference priors including the commonly used reference and “exact” reference priors for the SAR model. We show that common reference priors typically result in improper posterior distributions. Next, two “exact” reference priors are developed and are shown to yield proper posterior distributions. Frequentist properties of inferences based on two “exact” reference and Jeffreys-rule priors are studied by means of simulation. For illustrative purposes, we apply the method to SAT verbal scores across the US. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]