We introduce efficient Markov chain Monte Carlo methods for inference andmodel determination in multivariate and matrix-variate Gaussian graphicalmodels. Our framework is based on the G-Wishart prior for the precision matrixassociated with graphs that can be decomposable or non-decomposable. We extendour sampling algorithms to a novel class of conditionally autoregressive modelsfor sparse estimation in multivariate lattice data, with a special emphasis onthe analysis of spatial data. These models embed a great deal of flexibility inestimating both the correlation structure across outcomes and the spatialcorrelation structure, thereby allowing for adaptive smoothing and spatialautocorrelation parameters. Our methods are illustrated using simulated andreal-world examples, including an application to cancer mortality surveillance.
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