Variational Inference Aided Variable Selection For Spatially Structured High Dimensional Covariates

We consider the problem of Bayesian high dimensional variable selection in linear regression when a spatial structure exists among the covariates. We use an Ising prior to model the structural connectivity of the covariates with an undirected graph and the connectivity strength with Ising distribution parameters. Ising models, which originated in statistical physics, are widely used in computer vision and spatial data modeling. Although a Gibbs solution to this problem exists, the solution involves the computation of determinants and inverses of high dimensional matrices, rendering it unscalable to higher dimensions. Furthermore, the lack of theoretical support limits this important tool's use for the broader community. This paper proposes a variational inference-aided Gibbs approach that enjoys the same variable recovery power as the standard Gibbs solution while being computationally scalable to higher dimensions. We establish strong selection consistency for our proposed approach, along with its competitive numerical performance under varying simulation scenarios.