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图手术与 do-算子：无环结构因果模型的精确对应关系

文章背景与核心概要

本文探讨了 \(\operatorname{do}\)-算子在图形描述（删除指向目标的箭头）与函数实现（用常数替换目标机制）之间的形式等价性。作者针对具有有限内生变量的确定性无环结构因果模型，证明了两者在依赖层面的精确对应关系。

通过形式化机制依赖提取（\(\operatorname{Graph}(F)\)）和图手术（\(\operatorname{Surg}\)），核心定理表明，替换目标机制所消除的依赖关系与图手术所删除的依赖关系完全一致。此外，本文还刻画了图结构能够完美反映机制依赖的模型特征，定义了干预模型，追踪了执行过程，并证明了结果仅取决于实际依赖祖先节点的干预。

摘要

\(\operatorname{do}\)-算子在图形上表现为删除指向其目标的箭头，在函数上表现为用常数替换其机制。称这些操作为“等价”在数学上尚不严谨：前者返回一个图且仅记录目标，而后者返回机制并记录所施加的值。

我们针对具有有限内生变量的确定性无环结构因果模型，对依赖层面的比较进行了精确化。如果 \(\operatorname{Graph}(F)\) 提取了机制族 \(F\) 的依赖关系，我们的核心定理为：

The \(\operatorname{do}\)-operator is described graphically by deleting arrows into its targets and functionally by replacing their mechanisms with constants. To call these operations equivalent is not yet a mathematical statement: one returns a graph and remembers only the targets, whereas the other returns mechanisms and also remembers the imposed values.

We make a dependency-level comparison precise for deterministic acyclic structural causal models with finitely many endogenous variables. If \(\operatorname{Graph}(F)\) extracts the dependencies of a mechanism family \(F\), our main theorem is:

\[\operatorname{Graph}(F^\iota) = \operatorname{Surg}(\operatorname{Graph}(F), T_\iota)\]

因此，替换目标机制所消除的依赖关系与图手术所删除的依赖关系完全一致。对于一个图可能包含冗余箭头的模型 \(M = (G, F)\)，我们刻画了何时可以用 \(G\) 代替 \(\operatorname{Graph}(F)\) 成立该等式；当且仅当 \(G\) 精确记录了 \(F\) 的依赖关系时，该等式对每一次干预都成立。随后，我们定义了干预模型，刻画了其运行过程，展示了序贯干预如何组合，并证明了结果仅取决于其实际依赖祖先节点的干预。

Thus, replacing target mechanisms removes exactly the dependencies removed by graph surgery. For a model \(M = (G, F)\) whose graph may contain unused arrows, we characterize when the same equality holds with \(G\) in place of \(\operatorname{Graph}(F)\); it holds for every intervention exactly when \(G\) records the dependencies of \(F\) exactly. We then define the intervened model, characterize its run, show how sequential interventions combine, and prove that an outcome depends only on interventions at its actual dependency ancestors.

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