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Bayesian model selection approach for colored graphical Gaussian models

We consider a class of colored graphical Gaussian models obtained by placing symmetry constraints on the precision matrix in a Bayesian framework. The prior distribution on the precision matrix is the colored $G$-Wishart prior which is the Diaconis-Ylvisaker conjugate prior. In this paper, we develop a computationally efficient model search algorithm which combines linear regression with a double reversible jump Markov chain Monte Carlo (MCMC) method. The latter is to estimate the Bayes factors expressed as the ratio of posterior probabilities of two competing models. We also establish the asymptotic consistency property of the model selection procedure based on the Bayes factors. Our procedure avoids an exhaustive search which is computationally impossible. Our method is illustrated with simulations and a real-world application with a protein signalling data set.