  In many real-world scenarios, such as gene knockout experiments, targeted
interventions are often accompanied by unknown interventions at off-target
sites. Moreover, different units can get randomly exposed to different unknown
interventions, thereby creating a mixture of interventions. Identifying
different components of this mixture can be very valuable in some applications.
Motivated by such situations, in this work, we study the problem of identifying
all components present in a mixture of interventions on a given causal Bayesian
Network. We construct an example to show that, in general, the components are
not identifiable from the mixture distribution. Next, assuming that the given
network satisfies a positivity condition, we show that, if the set of mixture
components satisfy a mild exclusion assumption, then they can be uniquely
identified. Our proof gives an efficient algorithm to recover these targets
from the exponentially large search space of possible targets. In the more
realistic scenario, where distributions are given via finitely many samples, we
conduct a simulation study to analyze the performance of an algorithm derived
from our identifiability proof.
