Causal Inference Despite Limited Global Confounding via Mixture Models

Abstract

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

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Reproductions