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A Bayesian Network is a directed acyclic graph (DAG) on a set of $n$ randomvariables (the vertices); a Bayesian Network Distribution (BND) is aprobability distribution on the random variables that is Markovian on thegraph. A finite $k$-mixture of such models is graphically represented by alarger graph which has an additional ``hidden'' (or ``latent'') random variable$U$, ranging in $\{1,\ldots,k\}$, and a directed edge from $U$ to every othervertex. Models of this type are fundamental to causal inference, where $U$models an unobserved confounding effect of multiple populations, obscuring thecausal relationships in the observable DAG. By solving the mixture problem andrecovering the joint probability distribution with $U$, traditionallyunidentifiable causal relationships become identifiable. Using a reduction tothe more well-studied ``product'' case on empty graphs, we give the firstalgorithm to learn mixtures of non-empty DAGs.

Causal Inference Despite Limited Global Confounding via Mixture Models