Partial Identification of Counterfactual Distributions

This paper investigates the problem of bounding counterfactual queries from a combination of observational data and qualitative assumptions about the underlying data-generating model. These assumptions are usually represented in the form of a causal diagram (Pearl, 1995). We show that all counterfactual distributions (over finite observed variables) in an arbitrary causal diagram could be generated by a special family of structural causal models (SCMs), compatible with the same causal diagram, where unobserved (exogenous) variables are discrete, taking values in a finite domain. This entails a reduction in which the space where the original, arbitrary SCM lives can be mapped to a dual, more well-behaved space where the exogenous variables are discrete, and more easily parametrizable. Using this reduction, we translate the bounding problem in the original space into an equivalent optimization program in the new space. Solving such programs leads to optimal bounds over unknown counterfactuals. Finally, we develop effective Monte Carlo algorithms to approximate these optimal bounds from a finite number of observational data. Our algorithms are validated extensively on synthetic datasets.