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 we can 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.

翻译：我们假设给定一个由离散变量诱导的有向无环图结构方程，即结构因果模型，以及关于其内部节点的数据。我们想要回答的问题是：如何从这样的输入中计算部分可识别的反事实查询的界限。首先，我们给出从结构因果模型到信度网络的映射。这使得我们能够通过信度网络算法在结构因果模型的子类上精确计算反事实界限。然而，由于因果推断即使在多叉树上也是NP难问题，精确计算在一般情况下效率低下。因此，我们通过因果EM方案来近似计算界限。我们通过提供近似质量的置信区间来评估其准确性；并通过合成基准测试表明，该EM方案在相当数量的运行中能给出准确结果。在讨论过程中，我们还指出了一种常被忽视的局限性：即反事实界限无需结构方程知识即可计算这一流行观点。最后，我们通过一个关于姑息治疗的真实案例研究，展示了我们的算法如何直接应用于实际目的。