Bayesian model selection approach for coloured graphical Gaussian models

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