  Sparse Gaussian graphical models characterize sparse dependence relationships
between random variables in a network. To estimate multiple related Gaussian
graphical models on the same set of variables, we formulate a hierarchical
model, which leads to an optimization problem with a nonconvex log-shift
penalty function. We show that under mild conditions the optimization problem
is convex despite the inclusion of a nonconvex penalty, and derive an efficient
optimization algorithm. Experiments on both synthetic and real data show that
the proposed method is able to achieve good selection and estimation
performance simultaneously, because the nonconvexity of the log-shift penalty
allows for weak signals to be thresholded to zero without excessive shrinkage
on the strong signals.
