  Variable selection for structured covariates lying on an underlying known
graph is a problem motivated by practical applications, and has been a topic of
increasing interest. However, most of the existing methods may not be scalable
to high dimensional settings involving tens of thousands of variables lying on
known pathways such as the case in genomics studies. We propose an adaptive
Bayesian shrinkage approach which incorporates prior network information by
smoothing the shrinkage parameters for connected variables in the graph, so
that the corresponding coefficients have a similar degree of shrinkage. We fit
our model via a computationally efficient expectation maximization algorithm
which scalable to high dimensional settings (p~100,000). Theoretical properties
for fixed as well as increasing dimensions are established, even when the
number of variables increases faster than the sample size. We demonstrate the
advantages of our approach in terms of variable selection, prediction, and
computational scalability via a simulation study, and apply the method to a
cancer genomics study.
