Default Bayesian Analysis for Multivariate Generalized Car Models

In recent years, multivariate spatial models have been proven to be an effective tool for analyzing spatially related multidimensional data arising from a common underlying spatial process. The Bayesian analysis of these models is popular; however, the selection of an appropriate prior plays an important role in the inference. The two main contributions of this article are the development of shrinkage-type default priors for covariance matrices in these spatial models, and an innovative Gibbs sampling implementation that removes positive definiteness constraints when updating entries of the covariance matrix. The default prior elicitation is non-informative, but results in a proper posterior on the related parameter spaces. This elicitation not only provides robust inference (with respect to prior choice), but also provides improved estimation. In the computational step, the avoidance of sampling from restricted domains provides more stability and efficiency in the Gibbs implementation. Both simulations and data examples are provided to validate and illustrate the proposed methodology.