Efficient computation of counterfactual bounds

We assume to be given structural equations over discrete variables inducing a directed acyclic graph, namely, a structural causal model , together with data about its internal nodes. The question we want to answer is how can we compute bounds for partially identifiable counterfactual queries from such an input. We start by giving a map from structural casual models to credal networks . This allows us to compute exact counterfactual bounds via algorithms for credal nets on a subclass of structural causal models. Exact computation is going to be inefficient in general given that, as we show, causal inference is NP -hard even on polytrees. We target then approximate bounds via a causal EM scheme. We evaluate their accuracy by providing credible intervals on the quality of the approximation; we show through a synthetic benchmark that the EM scheme delivers accurate results in a fair number of runs. In the course of the discussion, we also point out what seems to be a neglected limitation to the trending idea that counterfactual bounds can be computed without knowledge of the structural equations. We also present a real case study on palliative care to show how our algorithms can readily be used for practical purposes.