Regression graphs and sparsity-inducing reparametrizations

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 is, under a causal ordering, a constrained version of the joint-response graphs studied by Cox and Wermuth (2004), while the most restrictive is that induced by sparsity on the scale of the matrix logarithm, studied by Battey (2017). This points to a class of reparametrizations for the chain-graph models Andersson et al. (2001), with undirected and directed acyclic graphs as special cases. While much of the paper is focused on exact zeros, the scope is considerably broadened through the possibility of approximate zeros. An important insight is the interpretation of these 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 other effects manifested through long paths. The insights have some conceptual implications; they also have a bearing on methodology, some aspects of which are developed in the supplementary material.