  Full Bayesian computational inference for model determination in undirected
graphical models is currently restricted to decomposable graphs, except for
problems of very small scale. In this paper we develop new, more efficient
methodology for such inference, by making two contributions to the
computational geometry of decomposable graphs. The first of these provides
sufficient conditions under which it is possible to completely connect two
disconnected complete subsets 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 numerical experiments on three
specific models.
