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，这些方程诱导出有向无环图（即结构因果模型），同时提供其内部节点的数据。本文要回答的问题是：如何从此类输入中计算部分可识别的反事实查询的边界？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针对结构因果模型子类的信度网络算法，精确计算反事实边界。然而，一般而言精确计算效率低下，因为正如我们所示，即使对于多叉树，因果推断也是NP难的。因此，我们通过因果期望最大化方案来近似计算边界。我们通过提供近似质量的置信区间来评估其准确性；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均能给出准确结果。在讨论过程中，我们还指出了一种似乎被忽视的局限：当前流行的观点认为无需了解结构方程即可计算反事实边界。最后，我们展示一项关于姑息治疗的现实案例研究，以说明所提算法如何可直接用于实际目的。