Coloured Gaussian directed acyclic graphical models

Abstract We study submodels of Gaussian directed acyclic graph (DAG) models defined by partial homogeneity constraints imposed on the model error variances and structural coefficients. We represent these models with coloured DAGs and investigate their properties for use in statistical and causal inference. Local and global Markov properties are provided and shown to characterize the coloured DAG model. Additional properties relevant to causal discovery are studied, including the existence and nonexistence of faithful distributions and structural identifiability. Extending prior work of Peters and Bühlmann 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 blocked properly edge-coloured DAGS (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 greedy equivalence search 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 coloured DAG models, coloured undirected graphical models and directed ancestral graph models. The proof yields a tool for identification of Markov properties for any rationally parametrized model with globally, rationally identifiable parameters.