Bayesian variable selection for correlated covariates via colored cliques

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We propose a Bayesian method to select groups of correlated explanatory variables in a linear regression framework. We do this by introducing in the prior distribution assigned to the regression coefficients a random matrix \(G\) that encodes the group structure. The groups can thus be inferred by sampling from the posterior distribution of \(G\). We then give a graph-theoretic interpretation of this random matrix \(G\) as the adjacency matrix of cliques. We discuss the extension of the groups from cliques to more general random graphs, so that the proposed approach can be viewed as a method to find networks of correlated covariates that are associated with the response.