Dis-entangling Mixture of Interventions on a Causal Bayesian Network Using Aggregate Observations

We study the problem of separating a mixture of distributions, all of which come from interventions on a known causal bayesian network. Given oracle access to marginals of all distributions resulting from interventions on the network, and estimates of marginals from the mixture distribution, we want to recover the mixing proportions of different mixture components. We show that in the worst case, mixing proportions cannot be identified using marginals only. If exact marginals of the mixture distribution were known, under a simple assumption of excluding a few distributions from the mixture, we show that the mixing proportions become identifiable. Our identifiability proof is constructive and gives an efficient algorithm recovering the mixing proportions exactly. When exact marginals are not available, we design an optimization framework to estimate the mixing proportions. Our problem is motivated from a real-world scenario of an e-commerce business, where multiple interventions occur at a given time, leading to deviations in expected metrics. We conduct experiments on the well known publicly available ALARM network and on a proprietary dataset from a large e-commerce company validating the performance of our method.