$\mathcal{G}$ which stipulates that the sets of potential outcome variables $\big\{ \{V_i(a_i) \mid a_i \in {\mathfrak X}_{\operatorname{pa}_{\mathcal{G}}(V_i)} \} \mid V_i \in V \big\}$ are mutually independent. All other counterfactuals may be defined using *recursive substitution*. For any $A \subseteq V \setminus \{ V_i \}$, $$\begin{aligned}
    V_i(a) \equiv V_i(a_{\operatorname{pa}_{\cal G}(V_i) \cap A},
    \{ V_j(a) : V_j \in \operatorname{pa}_{\cal G}(V_i) \setminus A \}),
    $$ where $\{ V_j(a) : V_j \in \operatorname{pa}_{\cal G}(V_i) \setminus A \}$ is taken to mean the (recursively defined) set of counterfactuals associated with variables in $\operatorname{pa}_{\cal G}(V_i) \setminus A$, had $A$ been set to $a$. Equivalently, Pearl’s model may be described by a system of nonparametric structural equations with independent errors.

A causal parameter is said to be *identified* in a causal model if it is a function of the observed data distribution $p(V)$. In the functional model of a DAG ${\cal G}$ (as well as some weaker causal models), all interventional distributions $p(V(a))$, for any $A \subseteq V$, are identified by the *extended g-formula*: $$p(V(a)) = \prod_{V_i \in V} \left. p(V_i | \operatorname{pa}_{\cal G}(V_i)) \right|_{A=a}.$$ For example, consider the DAG in Fig. \[fig:graphs\](a). $Y(a)$ is defined to be $Y(a,M(a,X),X)$ by recursive substitution and its distribution is identified 