Setup

• Let \(G=(V,E)\) be a DAG with vertex set \(V=\{1,\dots,m\}\).
• Let \(X=(X_1,\dots,X_m)\) have joint density \(f\) w.r.t.\ product measure \(\mu=\bigotimes_{v\in V}\mu_v\).

Assume the interventional distributions satisfy the causal Markov property for \(G\).

• Goal: Estimate the causal effect of \(X_T\) on \(X_R\) from observational data on \((X_T,X_R,X_C)\) with \(T,R,C\subseteq V\), \(T\cap R=\emptyset\)

Identification of Causal Effects

Suppose we observe \((X_T,X_R,X_C)\) for \(C\subseteq V\setminus(T\cup R)\).

1. When does covariate adjustment work? That is, when does
   \(f(x_R;\doOp(X_T=x_T)) \;=\; \int f(x_R\mid x_T,x_C)\,f(x_C)\,d\mu_C(x_C) \tag{A}\)
2. Is the causal effect identifiable? I.e.\ is \(f(x_R;\do(X_T=x_T))\) uniquely determined by the marginal of \((X_T,X_R,X_C)\)? :

Pearl’s Back-Door Criterion

Definition
A back-door path from \(t\) to \(r\) is any path beginning \(t\leftarrow \cdots \to r\).
A set \(C\subseteq V\setminus\{t,r\}\) satisfies the back-door criterion for \((t,r)\) if

1. \(C\cap \de_G(t)=\emptyset\),
2. every back-door path from \(t\) to \(r\) is d-blocked by \(C\).

Theorem 2
If \(C\) satisfies the back-door criterion for \((t,r)\), then the adjustment formula (A) holds.
Moreover, if \(C=\emptyset\),
\(f(x_r;\doOp(X_t=x_t)) \;=\; f(x_r\mid x_t).\) The back-door criterion is sufficient but not necessary for the adjustment formula to hold. A simple counter example is \(c\leftarrow t\rightarrow r\).

Theorem 3 (Adjustment criterion of Shpitser et al., 2012) Let \(C \subseteq V \setminus \{r, t\}\). The covariate adjustment formula (A) holds for all densities \(f\) that factorize according to the considered DAG \(G\) if and only if

• for all \(v \neq t\) that lie on a directed path from \(t\) to \(r\), we have \(C \cap \operatorname{de}(v) = \emptyset\);
• every path from \(t\) to \(r\) that is d-connecting given \(C\) is a directed path from \(t\) to \(r\).

Intervention Graphs

Definition
For each \(v\in A\subseteq V\), introduce an intervention node \(F_v\). The intervention graph has vertex set \(V\cup\{F_v:v\in A\}\) and edges \(E\cup\{F_v\to v:v\in A\}\).
Define the augmented conditionals
\(f'(x_v\mid x_{\pa(v)},F_v=i_v) = \begin{cases} f(x_v\mid x_{\pa(v)}),&i_v=\star,\\ \mathbf1\{x_v=i_v\},&i_v=x_v. \end{cases}\)
Key Fact: For any \(B\subseteq A\),
\(f(x;\do(X_B=x_B)) = f'\bigl(x\mid F_B=x_B^*,\;F_v=\emptyset, \; v\in A\setminus B \bigr).\)

Adjustment Criterion for Joint Interventions

Theorem 5 (Adjustment criterion)
Let \(T,R,C\subseteq V\) be pairwise disjoint. The covariate adjustment formula (A) holds for all densities \(f\) that factorize w.r.t. \(G\) if and only if:

1. For every \(v\notin T\) on a \(T\)-proper directed path to \(R\), no element of \(C\) is a descendant of \(v\) in the mutilated graph \(G_{\do(T)}\).
2. Every path from any \(t\in T\) to any \(r\in R\) that is d-connecting given \(C\) is either directed from \(T\) to \(R\) or is not \(T\)-proper.

Definition: A path is \(T\)-proper if its only node in \(T\) is its starting node.

With joint interventions, even when all variables in the given DAG are jointly observed, there might be no set, for which the adjustment formula works.

Efficiency of Estimators

In linear Gaussian SCMs, whenever (A) applies one can estimate the total effect of \(X_t\) on \(X_r\) by the regression coefficient of \(X_t\) in the linear regression of \(X_r\) on \((X_t,X_C)\). Different valid adjustment sets \(C\) yield estimators with different asymptotic variances; choosing a minimal or parent set often improves efficiency.