Bayesian Network distributions are fundamental to research in causal inference. We consider finite mixtures of such models, which are projections on the variables of a Bayesian Network distribution on the larger graph which has an additional hidden random variable U, ranging in {1, 2, ..., k}, and a directed edge from U to every other vertex. Thus, the confounding variable U selects the mixture constituent that determines the joint distribution of the observable variables. We give the first algorithm for identifying Bayesian Network distributions that can handle the case of non-empty graphs. The complexity for a graph of maximum degree ∆ (ignoring the degree of U) is roughly exponential in the number of mixture constituents k, and the degree ∆ squared (suppressing dependence on secondary parameters).