We study submodels of Gaussian DAG models defined by partial homogeneity constraints imposed on the model error variances and structural coefficients. We represent these models with colored DAGs and investigate their properties for use in statistical and causal inference. Local and global Markov properties are provided and shown to characterize the colored DAG model. Additional properties relevant to causal discovery are studied, including the existence and non-existence of faithful distributions and structural identifiability. Extending prior work of Peters and B\"uhlman and Wu and Drton, we prove structural identifiability under the assumption of homogeneous structural coefficients, as well as for a family of models with partially homogenous structural coefficients. The latter models, termed BPEC-DAGs, capture additional insights as they cluster the direct causes of each node into communities according to their effect on their common target. An analogue of the GES algorithm for learning BPEC-DAGs is given and evaluated on real and synthetic data. Regarding model geometry, we prove that these models are contractible, smooth, algebraic manifolds and compute their dimension. We also provide a proof of a conjecture of Sullivant which generalizes to colored DAG models, colored undirected graphical models and ancestral graph models.

翻译：我们研究由模型误差方差和结构系数的部分齐性约束所定义的高斯有向无环图（DAG）子模型。这些模型通过彩色DAG表示，并探究其在统计推断和因果推断中的应用特性。我们给出了局部和全局马尔可夫性质，并证明这些性质可刻画彩色DAG模型。同时研究了与因果发现相关的其他性质，包括忠实分布的存在性与非存在性，以及结构可识别性。我们拓展了Peters与Bühlmann以及Wu与Drton的前期工作，证明了在结构系数齐性假设下以及部分齐性结构系数模型族中的结构可识别性。后者被称为BPEC-DAG模型，通过将每个节点的直接原因按其对其同目标的效应划分为社群，从而捕获更多洞见。我们给出了用于学习BPEC-DAG的GES算法类似物，并在真实数据与合成数据上进行了评估。在模型几何方面，我们证明这些模型是可收缩的光滑代数流形，并计算了其维度。此外，我们提供了一个Sullivant猜想的证明，该猜想可推广至彩色DAG模型、彩色无向图模型和祖先图模型。