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：我们假设给定离散变量上的结构方程，这些方程诱导出一个有向无环图，即结构因果模型，同时附带关于其内部节点的数据。我们试图回答的问题是：如何从这样的输入中计算部分可识别的反事实查询的边界？我们首先给出从结构因果模型到置信网络的一个映射。这使我们能够通过针对结构因果模型子类的置信网络算法，精确计算反事实边界。然而，由于我们证明因果推断即使在多叉树上也是NP难的，因此在一般情况下精确计算将是低效的。接着，我们通过因果期望最大化方案来近似边界。我们通过提供近似质量的置信区间来评估其准确性；基于合成基准测试表明，该期望最大化方案在相当数量的运行中能交付准确结果。在讨论过程中，我们还指出，当前流行的“无需知道结构方程即可计算反事实边界”的观点似乎存在一个被忽视的局限性。此外，我们展示了一个关于姑息治疗的真实案例研究，以说明我们的算法如何能立即用于实际目的。