  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.
