Implementing a Metropolis sampler on decomposable graphs using a variety of ways to represent the graph

We describe the implementation of the Giudici-Green Metropolis sampling method for decomposable graphs using a variety of structures to represent the graph. These comprise the graph itself, the Junction tree, the Almond tree and the Ibarra clique-separator graph. For each structure, we describe the process for ascertaining whether adding or deleting a specific edge results in a new graph that is also decomposable, and the updates that need to be made to the structure if the edge perturbation is made. For the Almond tree and Ibarra graph these procedures are novel. We find that using the graph itself is generally at least competitive in terms of computational efficiency for a variety of graph distributions, but note that the other structures may allow and suggest samplers using different perturbations with lower rejection rates and/or better mixing properties. The sampler has applications in estimating graphical models for systems of multivariate Gaussian or Multinomial variables.