# Identifying Mixtures of Bayesian Network Distributions

A Bayesian Network is a directed acyclic graph (DAG) on a set of n random variables (identified with the vertices); a Bayesian Network Distribution (BND) is a probability distribution on the rv's that is Markovian on the graph. A finite mixture of such models is the projection on these variables of a BND on the 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 research in Causal Inference, where U models a confounding effect. One extremely special case has been of longstanding interest in the theory literature: the empty graph. Such a distribution is simply a mixture of k product distributions. A longstanding problem has been, given the joint distribution of a mixture of k product distributions, to identify each of the product distributions, and their mixture weights. Our results are: (1) We improve the sample complexity (and runtime) for identifying mixtures of k product distributions from exp(O(k^2)) to exp(O(k log k)). This is almost best possible in view of a known exp(Ω(k)) lower bound. (2) We give the first algorithm for the case of non-empty graphs. The complexity for a graph of maximum degree Δ is exp(O(k(Δ^2 + log k))). (The above complexities are approximate and suppress dependence on secondary parameters.)

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