Constructing priors based on model size for nondecomposable Gaussian graphical models: A simulation based approach

A method for constructing priors is proposed that allows the off-diagonal elements of the concentration matrix of Gaussian data to be zero. The priors have the property that the marginal prior distribution of the number of nonzero off-diagonal elements of the concentration matrix (referred to below as model size) can be specified flexibly. The priors have normalizing constants for each model size, rather than for each model, giving a tractable number of normalizing constants that need to be estimated. The article shows how to estimate the normalizing constants using Markov chain Monte Carlo simulation and supersedes the method of Wong et al. (2003) [24] because it is more accurate and more general. The method is applied to two examples. The first is a mixture of constrained Wisharts. The second is from Wong et al. (2003) [24] and decomposes the concentration matrix into a function of partial correlations and conditional variances using a mixture distribution on the matrix of partial correlations. The approach detects structural zeros in the concentration matrix and estimates the covariance matrix parsimoniously if the concentration matrix is sparse.