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.

翻译：参数化与稀疏性之间的内在联系表明，某些模型在其自然表述中虽不显式稀疏，但经过重参数化后可能在总体水平上呈现稀疏性。在协方差模型中，正定性对稀疏性的合理表现形式施加了额外约束。因此，自然需要考虑那些使稀疏性保持正定性的重参数化映射。本文的主要目的是深入探讨在物理自然尺度上，哪些结构会诱导重参数化后的稀疏性，以及稀疏性会诱导出哪些结构。最初发现的四种结构中最丰富的是——在因果序下——Cox和Wermuth（2004）研究的联合响应图的一个约束版本，而限制性最强的是由矩阵对数尺度上的稀疏性诱导的结构，由Battey（2017）研究。这指向了Andersson等人（2001）链图模型的一类重参数化方法，其中无向图和有向无环图作为特例。尽管本文大部分内容聚焦于精确零值，但通过近似零值的可能性，其适用范围得到了显著扩展。一个重要见解是对这些近似零值的解释，阐明了在重参数化后强制稀疏性的建模意义：实际上，如果两个变量之间的相对直接回归效应可忽略不计，而其他效应通过长路径显现，则它们之间的关系将被声明为空。这些见解具有一些概念性意义；它们也对方法论产生影响，其中某些方面在补充材料中有所发展。