On perfectness in Gaussian graphical models

Knowing when a graphical model perfectly encodes the conditional independence structure of a distribution is essential in applications, and this is particularly important when performing inference from data. When the model is perfect, there is a one-to-one correspondence between conditional independence statements in the distribution and separation statements in the graph. Previous work has shown that almost all models based on linear directed acyclic graphs as well as Gaussian chain graphs are perfect, the latter of which subsumes Gaussian graphical models (i.e., the undirected Gaussian models) as a special case. In this paper, we directly approach the problem of perfectness for the Gaussian graphical models, and provide a new proof, via a more transparent parametrization, that almost all such models are perfect. Our approach is based on, and substantially extends, a construction of Ln{\v{e}}ni{\v{c}}ka and Mat{\'{u}}{\v{s}} showing the existence of a perfect Gaussian distribution for any graph. The analysis involves constructing a probability measure on the set of normalized covariance matrices Markov with respect to a graph that may be of independent interest.