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 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 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. The proof yields a tool for the identification of Markov properties for rationally parameterized statistical models with globally, rationally identifiable parameters.

翻译：我们研究通过部分同质性约束施加于模型误差方差和结构系数所定义的高斯有向无环图模型的子模型。我们使用彩色有向无环图表示这些模型，并研究其在统计与因果推断中应用的特性。本文给出了局部与全局马尔可夫性质，并证明其刻画了彩色有向无环图模型。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。通过拓展Peters与Bühlman以及Wu与Drton的先前工作，我们证明了在同质结构系数假设下的结构可识别性，以及针对部分同质结构系数模型族的结构可识别性。后一类模型称为BPEC有向无环图，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见。本文给出了适用于学习BPEC有向无环图的GES算法改进版本，并在真实与合成数据上进行了评估。关于模型几何结构，我们证明这些模型是可收缩、光滑的代数流形，并计算了其维度。同时，我们提供了Sullivant猜想的证明，该猜想可推广至彩色有向无环图模型、彩色无向图模型及祖先图模型。该证明为具有全局有理可识别参数的有理参数化统计模型的马尔可夫性质识别提供了工具。