Sampling decomposable graphs using a Markov chain on junction trees

TL;DR

This paper introduces a new Markov chain Monte Carlo method for Bayesian inference in decomposable graphs, improving computational efficiency by leveraging geometric properties of junction trees.

Contribution

It provides new conditions for maintaining decomposability during graph modifications and develops a junction tree-based MCMC sampler for arbitrary distributions.

Findings

Efficient sampling of decomposable graphs demonstrated

New geometric conditions for graph connectivity established

Numerical experiments validate the methodology

Abstract

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.