Bayesian approaches for causal discovery can in principle quantify uncertainty in the prediction of the underlying causal structure, typically modeled by a directed acyclic graph (DAG). Various semi-implicit models for parametrized distributions over DAGs have been proposed, but their limitations have not been studied thoroughly. In this work, we focus on the expressiveness of parametrized distributions over DAGs in the context of causal structure learning and show several limitations of candidate models in a theoretical analysis and validate them empirically in supervised settings. To overcome them, we propose using mixture models of the considered distributions over DAGs.

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