  We study maximum likelihood estimation in Gaussian graphical models from a
geometric point of view. An algebraic elimination criterion allows us to find
exact lower bounds on the number of observations needed to ensure that the
maximum likelihood estimator (MLE) exists with probability one. This is applied
to bipartite graphs, grids and colored graphs. We also study the ML degree, and
we present the first instance of a graph for which the MLE exists with
probability one, even when the number of observations equals the treewidth.
