Identifiability

Definition
The causal effect \(P(X_R\in\cdot;\,\doOp(X_T=x_T))\) is identifiable from the marginal distribution of \(X_W\) if for any two observational distributions \(P^X_1, P^X_2\) with positive densities that factorize according to \(G\), \(P^X_{W,1} = P^X_{W,2} \quad\Longrightarrow\quad P_1\bigl(X_R\in\cdot;\,\doOp(X_T=x_T)\bigr) = P_2\bigl(X_R\in\cdot;\,\doOp(X_T=x_T)\bigr).\) In other words, an identifiable effect is uniquely determined by the marginal distribution of the observed variables \(X_W\).

Front–Door Criterion

Definition
Let \(r,t\in V\), \(r\neq t\). A set \(C\subseteq V\setminus\{r,t\}\) satisfies the front–door criterion with respect to the ordered pair \((t,r)\) if

1. every directed path from \(t\) to \(r\) contains a node in \(C\);
2. there is no back–door path from \(t\) to \(C\) that is d–connecting given \(\emptyset\);
3. there does not exist a back–door path from \(C\) to \(r\) that is d–connecting given \(\{t\}\).

Theorem
If \(C\) satisfies the front–door criterion wrt.\ \((t,r)\), then \(f\bigl(x_r;\,\doOp(X_t=x_t)\bigr) = \int f\bigl(x_C\mid x_t\bigr)\, \biggl[\int f\bigl(x_r\mid x_t,x_C\bigr)\,f(x_t)\,d\mu_t(x_t)\biggr] \,d\mu_C(x_C).\)

Do–Calculus

Notation:
\(A\perp B\mid C\) in \(G\) denotes that \(A\) and \(B\) are d-separated by \(C\) in the DAG \(G\).

Theorem (Rules of the Do–Calculus)
Let \(A,B,C,D\subseteq V\) be disjoint.

• Rule 1 (Insertion/deletion of observations):
  If \(A\perp B\mid (C,D)\) in \(G_C\), then
  \(f\bigl(x_A\mid x_B,x_D;\,\doOp(X_C=x_C)\bigr) = f\bigl(x_A\mid x_D;\,\doOp(X_C=x_C)\bigr).\)
• Rule 2 (Action/observation exchange):
  If \(A\perp B\mid (C,D)\) in \(G_{C,B}\), then
  \(f\bigl(x_A\mid x_D;\,\doOp(X_B=x_B,\,X_C=x_C)\bigr) = f\bigl(x_A\mid x_B,x_D;\,\doOp(X_C=x_C)\bigr).\)
• Rule 3 (Insertion/deletion of actions):
  Let \(B(D)=B\setminus\an_{G_C}(D)\). If \(A\perp B\mid (C,D)\) in \(G_{C\cup B(D)}\), then
  \(f\bigl(x_A\mid x_D;\,\doOp(X_B=x_B),\,\doOp(X_C=x_C)\bigr) = f\bigl(x_A\mid x_D;\,\doOp(X_C=x_C)\bigr).\)

Instrumental Variables

Consider the linear SCM \(\begin{aligned} X_1 &= \omega_{01} + \varepsilon_1,\\ X_2 &= \omega_{02} + \omega_{21}X_1 + \omega_{24}X_4 + \varepsilon_2,\\ X_3 &= \omega_{03} + \omega_{32}X_2 + \omega_{34}X_4 + \varepsilon_3,\\ X_4 &= \omega_{04} + \varepsilon_4, \end{aligned}\) with independent errors. By the trek rule, \(\Cov[X_1,X_2] = \omega_{21}\Var[X_1], \quad \Cov[X_1,X_3] = \omega_{32}\,\omega_{21}\Var[X_1].\) If \(\Cov[X_1,X_2]\neq 0\), then \(\omega_{32} = \frac{\Cov[X_1,X_3]}{\Cov[X_1,X_2]}.\) Hence the total effect \(\omega_{32}\) is (generically) identifiable from the distribution of \((X_1,X_2,X_3)\).