  Graphical Gaussian models have proven to be useful tools for exploring
network structures based on multivariate data. Applications to studies of gene
expression have generated substantial interest in these models, and resulting
recent progress includes the development of fitting methodology involving
penalization of the likelihood function. In this paper we advocate the use of
multivariate $t$-distributions for more robust inference of graphs. In
particular, we demonstrate that penalized likelihood inference combined with an
application of the EM algorithm provides a computationally efficient approach
to model selection in the $t$-distribution case. We consider two versions of
multivariate $t$-distributions, one of which requires the use of approximation
techniques. For this distribution, we describe a Markov chain Monte Carlo EM
algorithm based on a Gibbs sampler as well as a simple variational
approximation that makes the resulting method feasible in large problems.
