A SINful Approach to Model Selection for Gaussian Concentration Graphs

A multivariate Gaussian graphical Markov model for an undirected graph G, also called a covariance selection model or concentration graph model, is defined in terms of the Markov properties, i.e., conditional independences associated with G, which in turn are equivalent to specified zeroes among the set of pairwise partial correlation coefficients. By means of Fisher’s z-transformation and Šidák’s correlation inequality, conservative simultaneous confidence intervals for the entire set of partial correlations can be obtained, leading to a simple method for model selection that controls the overall error rate for incorrect edge inclusion. The simultaneous p-values corresponding to the partial correlations are partitioned into three disjoint sets, a significant set S, an indeterminate set I, and a non-significant set N. Our SIN model selection method selects two graphs, a graph ĜSI whose edges correspond to the set S ∪ I, and a more conservative graph ĜS whose edges correspond to S only. Prior information about the presence and/or absence of particular edges can be incorporated readily. Similar considerations apply to covariance graph models, which are defined in terms of marginal independence rather than conditional independence.