Recursive structural causal models on finite DAGs

Fix a finite DAG $G=(V,E)$ and a specified topological ordering $v_1,\dots ,v_n$ (every arrow points from an earlier to a later vertex). Partition $V=O\bigsqcup H$ into observed vertices $O$ and latent vertices $H$. For each $v\in V$, specify a measurable state space $(S_v,{\mathcal{S}}_v)$, a measurable exogenous space $(E_v,{\mathcal{E}}_v)$, a jointly measurable structural function $f_v:{\prod }_{u\in \mathrm{pa}(v)}{S}_u\times E_v\to S_v,$ and an exogenous random element $U_v$ with values in $E_v$. Here $\mathrm{pa}(v)=\{u:(u,v)\in E\}$; the empty product is a singleton. The model is Markovian when the finite family $(U_v{)}_{v\in V}$ is mutually independent. No parametric form or distributional truth is asserted. Correlated disturbances may be represented by adding their common causes as latent vertices, after which the exogenous family in the expanded model is required to be mutually independent.

Recursion in the topological order defines the endogenous random elements uniquely by $X_v=f_v(X_{\mathrm{pa}(v)},U_v)$. The full-joint law is the law of $(X_v{)}_{v\in V}$; the observed law is its marginal law on $O$. An observed-only DAG factorization is asserted only when $H=\mathrm{\varnothing }$ or when an additional stated latent-variable condition proves it.

For $I\subseteq V$ and a value $a=(a_v{)}_{v\in I}\in {\prod }_{v\in I}{S}_v$, $do(X_I=a)$ means replace only the equations indexed by $I$ by the constants $X_v=a_v$, leaving every other structural function and every exogenous law unchanged. Recursion then defines counterfactual values $X_w^a$ for every vertex $w$; the interventional law is their joint law. The response map $(a,\omega )\mapsto X_w^a(\omega )$ is jointly measurable, by induction through the finite topological order and joint measurability of each $f_v$. This permits evaluation at a random treatment value, such as $X_w^{M^a}$. When $I=\{A\}$, write $X_w^a$ for the response under $do(A=a)$. If the factual treatment is $A$, then on $\{A=a\}$ recursion gives $X_w=X_w^a$ for every vertex $w$.