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\"uhlmann 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 homogeneous structural coefficients. The latter models, termed BPEC-DAGs, capture additional causal insights by clustering 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 provide a proof of a conjecture of Sullivant which generalizes to colored DAG models, colored undirected graphical models and directed ancestral graph models. The proof yields a tool for identification of Markov properties for any rationally parameterized model with globally, rationally identifiable parameters.

翻译：我们研究了高斯有向无环图（DAG）模型的子模型，这些子模型通过对模型误差方差和结构系数施加部分同质性约束来定义。我们使用彩色有向无环图表示这些模型，并探讨它们在统计与因果推断中的应用特性。我们提供了局部和全局马尔可夫性质，并证明这些性质刻画了彩色有向无环图模型。我们还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。在Peters与Bühlmann以及Wu与Drton先前工作的基础上，我们证明了在结构系数同质化假设下的结构可识别性，以及针对一类具有部分同质结构系数模型的结构可识别性。后一类模型称为BPEC-DAG模型，通过根据节点对其共同目标的影响将其直接原因聚类为社群，从而捕捉额外的因果洞察。我们提出了适用于学习BPEC-DAG模型的GES算法变体，并在真实与合成数据上进行了评估。关于模型几何结构，我们证明了Sullivant的一个猜想，该猜想可推广至彩色有向无环图模型、彩色无向图模型以及有向祖先图模型。该证明为识别任何具有全局有理可识别参数的有理参数化模型的马尔可夫性质提供了工具。