  Graphical models are a framework for representing and exploiting prior
conditional independence structures within distributions using graphs. In the
Gaussian case, these models are directly related to the sparsity of the inverse
covariance (concentration) matrix and allow for improved covariance estimation
with lower computational complexity. We consider concentration estimation with
the mean-squared error (MSE) as the objective, in a special type of model known
as decomposable. This model includes, for example, the well known banded
structure and other cases encountered in practice. Our first contribution is
the derivation and analysis of the minimum variance unbiased estimator (MVUE)
in decomposable graphical models. We provide a simple closed form solution to
the MVUE and compare it with the classical maximum likelihood estimator (MLE)
in terms of performance and complexity. Next, we extend the celebrated Stein's
unbiased risk estimate (SURE) to graphical models. Using SURE, we prove that
the MSE of the MVUE is always smaller or equal to that of the biased MLE, and
that the MVUE itself is dominated by other approaches. In addition, we propose
the use of SURE as a constructive mechanism for deriving new covariance
estimators. Similarly to the classical MLE, all of our proposed estimators have
simple closed form solutions but result in a significant reduction in MSE.
