  Gaussian graphical models have received considerable attention during the
past four decades from the statistical and machine learning communities. In
Bayesian treatments of this model, the G-Wishart distribution serves as the
conjugate prior for inverse covariance matrices satisfying graphical
constraints. While it is straightforward to posit the unnormalized densities,
the normalizing constants of these distributions have been known only for
graphs that are chordal, or decomposable. Up until now, it was unknown whether
the normalizing constant for a general graph could be represented explicitly,
and a considerable body of computational literature emerged that attempted to
avoid this apparent intractability. We close this question by providing an
explicit representation of the G-Wishart normalizing constant for general
graphs.
