  Motivation: Recent advances in technology for brain imaging and
high-throughput genotyping have motivated studies examining the influence of
genetic variation on brain structure. Wang et al. (Bioinformatics, 2012) have
developed an approach for the analysis of imaging genomic studies using
penalized multi-task regression with regularization based on a novel group
$l_{2,1}$-norm penalty which encourages structured sparsity at both the gene
level and SNP level. While incorporating a number of useful features, the
proposed method only furnishes a point estimate of the regression coefficients;
techniques for conducting statistical inference are not provided. A new
Bayesian method is proposed here to overcome this limitation.
  Results: We develop a Bayesian hierarchical modeling formulation where the
posterior mode corresponds to the estimator proposed by Wang et al.
(Bioinformatics, 2012), and an approach that allows for full posterior
inference including the construction of interval estimates for the regression
parameters. We show that the proposed hierarchical model can be expressed as a
three-level Gaussian scale mixture and this representation facilitates the use
of a Gibbs sampling algorithm for posterior simulation. Simulation studies
demonstrate that the interval estimates obtained using our approach achieve
adequate coverage probabilities that outperform those obtained from the
nonparametric bootstrap. Our proposed methodology is applied to the analysis of
neuroimaging and genetic data collected as part of the Alzheimer's Disease
Neuroimaging Initiative (ADNI), and this analysis of the ADNI cohort
demonstrates clearly the value added of incorporating interval estimation
beyond only point estimation when relating SNPs to brain imaging
endophenotypes.
