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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$G$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>G</mi> </math> </EquationSource> </InlineEquation> that encodes the group structure. The groups can thus be inferred by sampling from the posterior distribution of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$G$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>G</mi> </math> </EquationSource> </InlineEquation>. We then give a graph-theoretic interpretation of this random matrix <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$G$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>G</mi> </math> </EquationSource> </InlineEquation> 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. Copyright Springer-Verlag Berlin Heidelberg 2014