The ratio of normalizing constants for Bayesian graphical Gaussian model selection
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The ratio of normalizing constants for the G-Wishart distribution, for two graphs differing by an edge e, has long been a bottleneck in the search for efficient model selection in the class of graphical Gaussian models. We give an accurate approximation to this ratio under two assumptions: first, we assume that the scale of the prior is the identity, second we assume that the set of paths between the two ends of e are disjoint. The first approximation does not represent a restriction since this is what statisticians use. The second assumption is a real restriction but we conjecture that similar results are also true without this second assumption. We shall prove it in subsequent work. This approximation is simply a ratio of Gamma functions and thus need no simulation. We illustrate the efficiency and practical impact of our result by comparing model selection in the class of graphical Gaussian models using this approximation and using current Metropolis-Hastings methods. We work both with simulated data and a complex high-dimensional real data set. In the numerical examples, we do not assume that the paths between the two endpoints of edge e are disjoint.
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