  Bayesian inference for graphical models has received much attention in the
literature in recent years. It is well known that when the graph G is
decomposable, Bayesian inference is significantly more tractable than in the
general non-decomposable setting. Penalized likelihood inference on the other
hand has made tremendous gains in the past few years in terms of scalability
and tractability. Bayesian inference, however, has not had the same level of
success, though a scalable Bayesian approach has its respective strengths,
especially in terms of quantifying uncertainty. To address this gap, we propose
a scalable and flexible novel Bayesian approach for estimation and model
selection in Gaussian undirected graphical models. We first develop a class of
generalized G-Wishart distributions with multiple shape parameters for an
arbitrary underlying graph. This class contains the G-Wishart distribution as a
special case. We then introduce the class of Generalized Bartlett (GB) graphs,
and derive an efficient Gibbs sampling algorithm to obtain posterior draws from
generalized G-Wishart distributions corresponding to a GB graph. The class of
Generalized Bartlett graphs contains the class of decomposable graphs as a
special case, but is substantially larger than the class of decomposable
graphs. We proceed to derive theoretical properties of the proposed Gibbs
sampler. We then demonstrate that the proposed Gibbs sampler is scalable to
significantly higher dimensional problems as compared to using an accept-reject
or a Metropolis-Hasting algorithm. Finally, we show the efficacy of the
proposed approach on simulated and real data.
