On rigorous specification of ICAR models

Intrinsic (or improper) conditional autoregressions, or ICARs, are widely used in spatial statistics, splines, dynamic linear models, and elsewhere. Such models usually have several variance components, including one for errors and at least one for random effects. Likelihood and Bayesian inference depend on the likelihood function of those variances. But in the absence of constraints or further specifications that are not inherent to ICARs, the likelihood function is arbitrary and thus so are some inferences. We suggest several ways to add constraints or further specifications, but any choice is merely a convention.