将因果概率推广到多值情形，给出可计算的闭式边界。
Identification of Probabilities of Causation: from Recursive to Closed-Form Bounds
- 基于结构因果模型，推导多值处理与结果的因果概率闭式边界。
- 闭式边界在低维情况下紧致，且优于现有递归边界。
- 适用于个性化决策与反事实分析，适合因果推断研究者。
因果概率（PoCs）是反事实分析与个性化决策的核心量。然而，现有理论主要局限于二值情形。本文通过在结构因果模型中利用标准实验与观测分布，将PoCs扩展至多值处理与结果，并为一类典型的离散PoCs推导出闭式边界。引入因果概率等价类概念，将任意离散PoCs简化为此类；建立可替换性原理，实现边界在值置换间的转移。所获边界在所有维度上均成立，低维情况下通过Balke线性规划方法验证其紧致性；进一步推测该紧致性在所有维度下均成立。模拟表明，闭式边界持续收紧近期递归边界，且计算更简单。最后，通过小规模示例展示其实际意义。
原文摘要 · Abstract (English)
Probabilities of causation (PoCs) are fundamental quantities for counterfactual analysis and personalized decision making. However, existing analytical results are largely confined to binary settings. This paper extends PoCs to multi-valued treatments and outcomes by deriving closed form bounds for a representative family of discrete PoCs within Structural Causal Models, using standard experimental and observational distributions. We introduce the notion of equivalence classes of PoCs, which reduces arbitrary discrete PoCs to this family, and establish a replaceability principle that transfers bounds across value permutations. For the resulting bounds, we prove soundness in all dimensions and empirically verify tightness in low dimensional cases via Balke's linear programming method; we further conjecture that this tightness extends to all dimensions. Simulations indicate that our closed form bounds consistently tighten recent recursive bounds while remaining simpler to compute. Finally, we illustrate the practical relevance of our results through toy examples.
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