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-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.
Hodges
James S
JS
Division of Biostatistics, School of Public Health, University of Minnesota, MMC 303, 420 Delaware St. SE, Minneapolis, Minnesota 55455, USA. hodges@ccbr.umn.edu
Carlin
Bradley P
BP
Fan
Qiao
Q
D001499:Bayes Theorem; D003627:Data Interpretation, Statistical; D006801:Humans; D008390:Markov Chains; D015233:Models, Statistical; D009010:Monte Carlo Method; D017622:Periodontal Attachment Loss / Q000473:pathology; D010518:Periodontitis / Q000188:drug therapy