On the Precision of the Conditionally Autoregressive Prior in Spatial Models

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 τ k for some k . This article rigorously derives k = ( n − I )/2 for the L 2 norm CAR prior (also called a Gaussian Markov random field model) and k = n − I for the L 1 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 L 2 norm case, instead of the more common k = n /2 . We illustrate the practical significance of our results using a periodontal example.