Sampling decomposable graphs using a Markov chain on junction trees

This paper makes two contributions to the computational geometry of decomposable graphs, aimed primarily at facilitating statistical inference about such graphs where they arise as assumed conditional independence structures in stochastic models. The first of these provides sufficient conditions under which it is possible to completely connect two disconnected cliques of vertices, or perform the reverse procedure, yet maintain decomposability of the graph. The second is a new Markov chain Monte Carlo sampler for arbitrary positive distributions on decomposable graphs, taking a junction tree representing the graph as its state variable. The resulting methodology is illustrated with two numerical experiments.