On the precision of the conditionally autoregressive prior in spatial models

James S Hodges, Brad Carlin, Qiao Fan

Research output: Contribution to journalArticlepeer-review

57 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)317-322
Number of pages6
Issue number2
StatePublished - Jun 2003


  • Bayesian analysis
  • Gaussian Markov random field model
  • Improper prior
  • Markov chain Monte Carlo (MCMC) methods
  • Periodontal data


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