  We investigate the choice of tuning parameters for a Bayesian multi-level
group lasso model developed for the joint analysis of neuroimaging and genetic
data. The regression model we consider relates multivariate phenotypes
consisting of brain summary measures (volumetric and cortical thickness values)
to single nucleotide polymorphism (SNPs) data and imposes penalization at two
nested levels, the first corresponding to genes and the second corresponding to
SNPs. Associated with each level in the penalty is a tuning parameter which
corresponds to a hyperparameter in the hierarchical Bayesian formulation.
Following previous work on Bayesian lassos we consider the estimation of tuning
parameters through either hierarchical Bayes based on hyperpriors and Gibbs
sampling or through empirical Bayes based on maximizing the marginal likelihood
using a Monte Carlo EM algorithm. For the specific model under consideration we
find that these approaches can lead to severe overshrinkage of the regression
parameter estimates in the high-dimensional setting or when the genetic effects
are weak. We demonstrate these problems through simulation examples and study
an approximation to the marginal likelihood which sheds light on the cause of
this problem. We then suggest an alternative approach based on the widely
applicable information criterion (WAIC), an asymptotic approximation to
leave-one-out cross-validation that can be computed conveniently within an MCMC
framework.
