Decomposable graphical gaussian model determination
by Paolo Giudici (Universita' di Pavia)
and Peter J. Green (University of Bristol)
We propose a methodology for Bayesian model determination in
decomposable graphical gaussian models. To achieve this aim we consider
a hyper inverse Wishart prior distribution on the concentration matrix
for each given graph. To ensure compatibility across models, such prior
distributions are obtained by marginalisation from the prior
conditional on the complete graph. We explore alternative structures
for the hyperparameters of the latter, and their consequences for the
model. Model determination is carried out by implementing a reversible
jump MCMC sampler. In particular, the dimension-changing move we
propose involves adding or dropping an edge from the graph. We
characterise the set of moves which preserve the decomposability of the
graph, giving a fast algorithm for maintaining the junction tree
representation of the graph at each sweep. As state variable, we
propose to use the incomplete variance-covariance matrix, containing
only the elements for which the corresponding element of the inverse is
nonzero. This allows all computations to be performed locally, at the
clique level, which is a clear advantage for the analysis of large and
complex data-sets. Finally, the statistical and computational
performance of the procedure is illustrated by means of both artificial
and real data-sets.
Some key words:
Bayesian Model Selection; Hyper Markov distributions;
Junction Tree; Inverse Wishart Distribution; Reversible Jump MCMC.
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