Unifying methodologies for graphical models with Gaussian parametrization

Graphical models represent conditional independences of a multivariate distribution by absent edges in a graph, which typically is directed, undirected or mixed. This compact modelling allows to decompose statistical inference into efficient computations over the associated graph. As such, graphical models originated mainly at the interface between statistics and artificial intelligence, with Markov networks (undirected graph) and Bayesian networks (acyclic digraph) being the classic representatives. Nowadays graphical models are widely applied and a significant amount of research is devoted to them. Gaussian Markov networks and Gaussian Bayesian networks, although not being equivalent models, share a common intersection consisting of chordal graphs (or acyclic digraphs with no v-structures). A typical approach for model selection in both model classes is hypothesis testing, which amounts to selecting the graph that parametrizes the model. Absent edges in both models are represented by a zero pattern in the inverse covariance or partial correlation matrix (Gaussian Markov networks) or in its Cholesky decomposition (Gaussian Bayesian networks). Afterwards, their parameters are estimated by maximum likelihood. Alternatively, there exist state-of-the-art regularisation methods for both model classes, which simultaneously perform model selection and estimation. A popular method for model selection via hypothesis testing is the PC algorithm, which can be applied for both Gaussian Markov networks and Gaussian Bayesian networks. This method mainly depends on two parameters: the statistical test type and the significance level at which the hypotheses are tested. However, the usual approach in the literature is to use a Gaussian test for a transformation of the partial correlation, and a grid search for its significance level. By contrast, when using an automatic procedure for parameter tuning, such as Bayesian optimization, it is shown how model selection performance is significantly improved when employing an uncommonly used test in the literature. Furthermore, these automatic parameter tuning procedures allow to select a significance level optimized for each test type. Validation of methodologies for Gaussian graphical model selection is also deeply affected, apart from how parameter tuning is performed, by how synthetic test models are simulated. It can be shown that state-of-the-art methodologies addressing this task are biased towards certain regions, thereby significantly influencing validation results. It would be therefore desirable to have a uniform sampling procedure for Gaussian graphical models. In particular, both Gaussian Bayesian networks and Gaussian Markov networks are intimately related with the partial correlation matrix, thereby uniform sampling methods for such set, called elliptope, can be a departing point. A novel Metropolis uniform sampling from the elliptope is proposed, which can be straightforwardly extended to chordal Gaussian graphical models. However, in the general case, a partial orthogonalization method has to be used for Gaussian Markov networks, and the results are not