Regression graphs and sparsity-inducing reparameterizations

Summary That parameterization and sparsity are inherently linked raises the possibility that relevant models, not obviously sparse in their natural formulation, exhibit a population-level sparsity after reparameterization. In covariance models, positive definiteness enforces additional constraints on how sparsity can legitimately manifest. It is therefore natural to consider reparameterization maps in which sparsity respects positive definiteness. This paper provides insight into structures on the physically natural scale that induce and are induced by sparsity after reparameterization. Of the four structures initially uncovered, the richest can be generated, under a causal ordering, by the joint-response graphs studied by Wermuth & Cox (2004). This connection leads to an interpretation of approximate zeros and explains modelling implications of enforcing sparsity after reparameterization: in effect, the relation between two variables would be declared null if relatively direct regression effects were negligible and other effects manifested through long paths. The Iwasawa decomposition of the general linear group, combined with the graphical-model interpretation, points to a class of reparameterizations for the chain-graph models (Andersson et al., 2001), with undirected and directed acyclic graphs as special cases. The insights have a bearing on methodology, some aspects of which are developed. An extensive simulation uses the theoretical insights to further explore regimes under which reparameterization is beneficial.