Learning to Bound Counterfactual Inference in Structural Causal Models from Observational and Randomised Data

We address the problem of integrating data from multiple observational and interventional studies to eventually compute counterfactuals in structural causal models. We derive a likelihood characterisation for the overall data that leads us to extend a previous EM-based algorithm from the case of a single study to that of multiple ones. The new algorithm learns to approximate the (unidentifiability) region of model parameters from such mixed data sources. On this basis, it delivers interval approximations to counterfactual results, which collapse to points in the identifiable case. The algorithm is very general, it works on semi-Markovian models with discrete variables and can compute any counterfactual. Moreover, it automatically determines if a problem is feasible (the parameter region being nonempty), which is a necessary step not to yield incorrect results. Systematic numerical experiments show the effectiveness and accuracy of the algorithm, while hinting at the benefits of integrating heterogeneous data to get informative bounds in case of unidentifiability.