Regression graphs and sparsity-inducing reparametrizations

Abstract

That parametrization and sparsity are inherently linked raises the possibility that relevant models, not obviously sparse in their natural formulation, exhibit a population-level sparsity after reparametrization. In covariance models, positive-definiteness enforces additional constraints on how sparsity can legitimately manifest. It is therefore natural to consider reparametrization maps in which sparsity respects positive definiteness. The main purpose of this paper is to provide insight into structures on the physically-natural scale that induce and are induced by sparsity after reparametrization. The richest of the four structures initially uncovered can be generated, under a causal ordering, by the joint-response graphs studied by Wermuth & Cox (2004), while the most restrictive is that induced by sparsity on the scale of the matrix logarithm, studied by Battey (2017). The Iwasawa decomposition of the general linear group, combined with the graphical-models interpretation, points to a class of reparametrizations for the chain-graph models (Andersson et al. 2001), with undirected and directed acyclic graphs as special cases. An important insight is the interpretation of approximate zeros, explaining the 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 others manifested through long paths. The insights have a bearing on methodology, some aspects of which are presented. A detailed simulation uses the theoretical insights to further explore regimes under which reparametrization is beneficial.