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. In the talk I will aim to provide insight into structures on the physically-natural scale that induce and are induced by sparsity after reparametrization. The most restrictive of the four structures initially uncovered is that induced by sparsity on the scale of the matrix logarithm, studied by Battey (2017), while the richest is interpretable in terms of the joint-response graphs studied by Wermuth & Cox (2004). This points to a class of reparametrizations for the chain-graph models (Andersson et al., 2001) when a causal ordering is present, 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 talk is based on joint work with Jakub Rybak and Karthik Bharath.