  Gene-environment (G$\times$E) interactions have important implications to
elucidate the etiology of complex diseases beyond the main genetic and
environmental effects. Outliers and data contamination in disease phenotypes of
G$\times$E studies have been commonly encountered, leading to the development
of a broad spectrum of robust regularization methods. Nevertheless, within the
Bayesian framework, the issue has not been taken care of in existing studies.
We develop a fully Bayesian robust variable selection method for G$\times$E
interaction studies. The proposed Bayesian method can effectively accommodate
heavy-tailed errors and outliers in the response variable while conducting
variable selection by accounting for structural sparsity. In particular, for
the robust sparse group selection, the spike-and-slab priors have been imposed
on both individual and group levels to identify important main and interaction
effects robustly. An efficient Gibbs sampler has been developed to facilitate
fast computation. Extensive simulation studies and analysis of both the
diabetes data with SNP measurements from the Nurses' Health Study and TCGA
melanoma data with gene expression measurements demonstrate the superior
performance of the proposed method over multiple competing alternatives.
