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 (tau) that is typically unknown. To include (tau) in the Bayesian analysis, the kernel must be multiplied by (tau)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-Heldʹs (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.