The problem of finding an ancestral acyclic directed mixed graph (ADMG) that represents the causal relationships between a set of variables is an important area of research for causal inference. However, most of existing score-based structure learning methods focus on learning the directed acyclic graph (DAG) without latent variables. A number of score-based methods have recently been proposed for the ADMG learning, yet they are heuristic in nature and do not guarantee an optimal solution. We propose a novel exact score-based method that solves an integer programming (IP) formulation and returns a score-maximizing ancestral ADMG for a set of continuous variables. In particular, we generalize the state-of-the-art IP model for DAG learning problems and derive new classes of valid inequalities to formalize the IP-based ADMG learning model. Empirically our model can be solved efficiently for medium-sized problems and achieves better accuracy than state-of-the-art score-based methods as well as benchmark constraint-based methods.
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