Directed acyclic graphs (DAGs) are commonly used to model causal relationships among random variables. In general, learning the DAG structure is both computationally and statistically challenging. Moreover, without additional information, the direction of edges may not be estimable from observational data. In contrast, given a complete causal ordering of the variables, the problem can be solved efficiently, even in high dimensions. In this paper, we consider the intermediate problem of learning DAGs when a partial causal ordering of variables is available. We propose a general estimation framework for leveraging the partial ordering and present efficient estimation algorithms for low- and high-dimensional problems. The advantages of the proposed framework are illustrated via numerical studies.

翻译：有向无环图（DAG）常被用于建模随机变量间的因果关系。通常，学习DAG结构在计算和统计上均具有挑战性。此外，若无额外信息，边的方向可能无法从观测数据中估计。相比之下，在给定变量完整因果排序的情况下，即使在高维场景下也能高效求解该问题。本文考虑可获取变量部分因果排序时的DAG学习这一中间问题。我们提出一个利用部分排序的通用估计框架，并针对低维和高维问题分别开发了高效估计算法。通过数值实验验证了所提框架的优势。