A graph structure is commonly used to characterize the dependence between variables, which may be induced by time, space, biological networks or other factors. Incorporating this dependence structure into the variable selection procedure can improve the identification of relevant variables, especially those with subtle effects. The Bayesian approach provides a natural framework to integrate this information through the prior distributions. In this work, we propose combining two priors that have been well studied separately-the Gaussian Markov random field prior and the horseshoe prior-to perform selection on graph-structured variables. Local shrinkage parameters that capture the dependence between connected covariates are specified to encourage similar amount of shrinkage for their regression coefficients, while a standard horseshoe prior is used for non-connected variables. After evaluating the performance of the method on different simulated scenarios, we present three applications: one in quantitative trait loci mapping with block sequential structure, one in near-infrared spectroscopy with sequential non-disjoint dependence and another in gene expression study with a general dependence structure.
