Gaussian Markov Random Fields: Theory And Applications (Monographs on Statistics and Applied Probability)
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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.