Disentangling Causal Effects from Sets of Interventions in the Presence of Unobserved Confounders

The ability to answer causal questions is crucial in many domains, as causal inference allows one to understand the impact of interventions. In many applications, only a single intervention is possible at a given time. However, in certain important areas, multiple interventions are concurrently applied. Disentangling the effects of single interventions from jointly applied interventions is a challenging task---especially as simultaneously applied interventions can interact. This problem is made harder still by unobserved confounders, which influence both interventions and outcome. We address this challenge by aiming to learn the effect of a single-intervention from both observational data and sets of interventions. We prove that this is not generally possible, but provide identification proofs demonstrating that it can be achieved in certain classes of additive noise models---even in the presence of unobserved confounders. Importantly, we show how to incorporate observed covariates and learn heterogeneous treatment effects conditioned on them for single-interventions.