Regression graphs and sparsity-inducing reparametrizations

That parametrization and population-level sparsity are intrinsically 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. In a sense the richest of the four structures initially uncovered turns out to be that of 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). 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 after reparametrization, an important insight is the interpretation of approximate zeros, which explains 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 a bearing on methodology, some aspects of which are discussed in the supplementary material where an estimator with high-dimensional statistical guarantees is presented.