Title: The ratio of normalizing constants for Bayesian graphical Gaussian model selection,
Abstract: Many graphical Gaussian selection methods in a Bayesian framework use the
G-Wishart as the conjugate prior on the precision matrix. The Bayes factor to
compare a model governed by a graph G and a model governed by the neighboring
graph G-e, derived from G by deleting an edge e, is a function of the ratios of
prior and posterior normalizing constants of the G-Wishart for G and G-e.
While more recent methods avoid the computation of the posterior ratio,
computing the ratio of prior normalizing constants, (2) below, has remained a
computational stumbling block. In this paper, we propose an explicit analytic
approximation to (2) which is equal to the ratio of two Gamma functions
evaluated at (delta+d)/2 and (delta+d+1)/2 respectively, where delta is the
shape parameter of the G-Wishart and d is the number of paths of length two
between the endpoints of e. This approximation allows us to avoid Monte Carlo
methods, is computationally inexpensive and is scalable to high-dimensional
problems. We show that the ratio of the approximation to the true value is
always between zero and one and so, one cannot incur wild errors.
In the particular case where the paths between the endpoints of e are
disjoint, we show that the approximation is very good. When the paths between
these two endpoints are not disjoint we give a sufficient condition for the
approximation to be good. Numerical results show that the ratio of the
approximation to the true value of the prior ratio is always between .55 and 1
and very often close to 1. We compare the results obtained with a model search
using our approximation and a search using the double Metropolis-Hastings
algorithm to compute the prior ratio. The results are extremely close.