On the precision of the conditionally autoregressive prior in spatial models

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 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.