On the Precision of the Conditionally Autoregressive Prior in Spatial Models

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Abstract
Summary Bayesian analyses of spatial data often use a conditionally autoregressive (CAR) prior, which can be written as the kernel of an improper density that depends on a precision parameter τ that is typically unknown. To include τ in the Bayesian analysis, the kernel must be multiplied by τk for some k. This article rigorously derives k= (n−I)/2 for the L2 norm CAR prior (also called a Gaussian Markov random field model) and k=n−I for the L1 norm CAR prior, where n is the number of regions and I the number of “islands” (disconnected groups of regions) in the spatial map. Since I= 1 for a spatial structure defining a connected graph, this supports Knorr‐Held's (2002, in Highly Structured Stochastic Systems, 260–264) suggestion that k= (n− 1)/2 in the L2 norm case, instead of the more common k=n/2. We illustrate the practical significance of our results using a periodontal example.

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