Partial Counterfactual Identification from Observational and
  Experimental Data
  This paper investigates the problem of bounding counterfactual queries from
an arbitrary collection of observational and experimental distributions and
qualitative knowledge about the underlying data-generating model represented in
the form of a causal diagram. We show that all counterfactual distributions in
an arbitrary structural causal model (SCM) could be generated by a canonical
family of SCMs with the same causal diagram where unobserved (exogenous)
variables are discrete with a finite domain. Utilizing the canonical SCMs, we
translate the problem of bounding counterfactuals into that of polynomial
programming whose solution provides optimal bounds for the counterfactual
query. Solving such polynomial programs is in general computationally
expensive. We therefore develop effective Monte Carlo algorithms to approximate
the optimal bounds from an arbitrary combination of observational and
experimental data. Our algorithms are validated extensively on synthetic and
real-world datasets.
