Bayesian variable selection for correlated covariates via colored cliques

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.