Representing Sparse Gaussian DAGs as Sparse R-Vines Allowing for Non-Gaussian Dependence

Modeling dependence in high-dimensional systems has become an increasingly important topic. Most approaches rely on the assumption of a multivariate Gaussian distribution such as statistical models on directed acyclic graphs (DAGs). They are based on modeling conditional independencies and are scalable to high dimensions. In contrast, vine copula models accommodate more elaborate features like tail dependence and asymmetry, as well as independent modeling of the marginals. This flexibility comes however at the cost of exponentially increasing complexity for model selection and estimation. We show a novel connection between DAGs with limited number of parents and truncated vine copulas under sufficient conditions. This motivates a more general procedure exploiting the fast model selection and estimation of sparse DAGs while allowing for non-Gaussian dependence using vine copulas. By numerical examples in hundreds of dimensions, we demonstrate that our approach outperforms the standard method for vine structure selection. Supplementary material for this article is available online.

高维系统中的依赖关系建模已成为愈发重要的研究课题。现有多数方法均基于多元高斯分布假设，例如基于有向无环图（directed acyclic graphs, DAGs）的统计模型。这类方法以条件独立性建模为核心，且可扩展至高维场景。与之相比，藤Copula（vine copula）模型能够适配更精细的特征，例如尾部相关性与非对称性，同时支持边缘分布的独立建模。但这类灵活性的代价是模型选择与参数估计的复杂度呈指数级增长。本文证明了，在充分条件下，父节点数量受限的有向无环图与截断型藤Copula模型之间存在全新关联。这一发现催生了一种更通用的建模流程：该流程可利用稀疏有向无环图的快速模型选择与参数估计能力，同时借助藤Copula模型实现非高斯依赖关系的建模。通过数百维场景下的数值实验，本文证明所提方法的表现优于藤结构选择的标准方法。本文补充材料可在线获取。