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Sampling decomposable graphs using a Markov chain on junction trees

by Peter Green and Alun Thomas

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.

Key words: conditional independence graph, graphical model, Markov chain Monte Carlo, Markov random field, model determination.

Full text of the paper, which has been accepted by Biometrika.