We consider the problem of learning a Gaussian graphical model in the case where the observations come from two dependent groups sharing the same variables. We focus on a family of coloured Gaussian graphical models specifically suited for the paired data problem. Commonly, graphical models are ordered by the submodel relationship so that the search space is a lattice, called the model inclusion lattice. We introduce a novel order between models, named the twin order. We show that, embedded with this order, the model space is a lattice that, unlike the model inclusion lattice, is distributive. Furthermore, we provide the relevant rules for the computation of the neighbours of a model. The latter are more efficient than the same operations in the model inclusion lattice, and are then exploited to achieve a more efficient exploration of the search space. These results can be applied to improve the efficiency of both greedy and Bayesian model search procedures. Here, we implement a stepwise backward elimination procedure and evaluate its performance both on synthetic and real-world data.

Learning of large-scale networks of interactions from microarray data is an important and challenging problem in bioinformatics. A widely used approach is to assume that the available data constitute a random sample from a multivariate distribution belonging to a Gaussian graphical model. As a consequence, the prime objects of inference are full-order partial correlations which are partial correlations between two variables given the remaining ones. In the context of microarray data the number of variables exceed the sample size and this precludes the application of traditional structure learning procedures because a sampling version of full-order partial correlations does not exist. In this paper we consider limited-order partial correlations, these are partial correlations computed on marginal distributions of manageable size, and provide a set of rules that allow one to assess the usefulness of these quantities to derive the independence structure of the underlying Gaussian graphical model. Furthermore, we introduce a novel structure learning procedure based on a quantity, obtained from limited-order partial correlations, that we call the non-rejection rate. The applicability and usefulness of the procedure are demonstrated by both simulated and real data.

This paper deals with chain graph models under alternative AMP interpretation. A new representative of an AMP Markov equivalence class, called the largest deflagged graph, is proposed. The representative is based on revealed internal structure of the AMP Markov equivalence class. More specifically, the AMP Markov equivalence class decomposes into finer strong equivalence classes and there exists a distinguished strong equivalence class among those forming the AMP Markov equivalence class. The largest deflagged graph is the largest chain graph in that distinguished strong equivalence class. A composed graphical procedure to get the largest deflagged graph on the basis of any AMP Markov equivalent chain graph is presented. In general, the largest deflagged graph differs from the AMP essential graph, which is another representative of the AMP Markov equivalence class.