Setup

Let \(G=(V,E)\) be a DAG and assume the causal Markov property so that \(f(x)=\prod_{v\in V}f(x_v\mid x_{\pa(v)}),\quad f(x)>0.\)

Interventional Distributions

For \(A\subseteq V\), the interventional density under \(\doOp(X_A=x_A)\) is \(f\bigl(x;\,\doOp(X_A=x_A)\bigr) =\prod_{v\notin A}f(x_v\mid x_{\pa(v)})\;\times\;\prod_{v\in A}\mathbf1\{x_v=x_A\}.\)

Causal Effect

Definition
Let \(T,R\subseteq V\). The causal effect of \(X_T\) on \(X_R\) is the map \(x_T\;\mapsto\;P\bigl(X_R\in\cdot\;;\,\doOp(X_T=x_T)\bigr).\) Often for \(\lvert T\rvert=\lvert R\rvert=1\) one considers:

• If \(X_T\) is binary, the average treatment effect \(\;E[X_R;\doOp(X_T=1)] - E[X_R;\doOp(X_T=0)].\)
• In a linear SCM, \(E[X_R;\doOp(X_T=x_T)]=\omega_0+\omega x_T\), and \(\omega\) is the total effect.

Total Effects in Linear Structural Causal Models

Consider the linear SEM \(X_v = \omega_{0v} + \sum_{u\in\pa(v)}\omega_{v u}\,X_u + \vartheta_v,\quad v\in V,\) where \(\{\vartheta_v\}\) are independent mean-zero errors. Set \(B=(\omega_{v u})\) and let the path matrix be \((I-B)^{-1}\).

Example
For \(V=\{1,2,3\}\) with \(X_1=\omega_{01}+\vartheta_1,\quad X_2=\omega_{02}+\omega_{21}X_1+\vartheta_2,\quad X_3=\omega_{03}+\omega_{31}X_1+\omega_{32}X_2+\vartheta_3,\) the total effects \(C(t,r)\) are: \(C(1,2)=\omega_{21},\; C(1,3)=\omega_{31}+\omega_{32}\,\omega_{21},\; C(2,3)=\omega_{32},\) and zero for reverse ordering.

Proposition
In this linear SCM, the total effect of \(X_t\) on \(X_r\) is \(C(t,r)\;=\;(I-B)^{-1}_{r t} \;=\; \sum_{\pi\in\Pi(t,r)}\;\prod_{(v\to w)\in\pi}\beta_{w v},\) where \(\Pi(t,r)\) is the set of directed paths from \(t\) to \(r\).

Adjust for Source Nodes

Remark Causal effect of Xt on XR depends on the conditional distribution of X_R\mid X_t

Adjusting for Parents

Theorem
Let \(t\in V\) and \(R\subseteq V\setminus(\{t\}\cup\pa(t))\). Then \(f\bigl(x_R;\,\doOp(X_t=x_t)\bigr) =\int f\bigl(x_R\mid x_t,x_{\pa(t)}\bigr)\, f\bigl(x_{\pa(t)}\bigr)\, d\mu_{\pa(t)}(x_{\pa(t)}).\)

Remark
This shows the causal effect of \(X_t\) on \(X_R\) is identified by the marginal law of \((X_t,X_{\pa(t)},X_R)\).

Identification of Causal Effects

Suppose we observe the vairables \(X_T\), \(X_R\) and \(X_C\) for a set \(C \subseteq V \setminus [T \cup R].\)

1. When does covariate adjustment work? That is, when do we have \(f(x_R; \doOp(X_T = x_T^*)) = \int f(x_R \mid x_T^*, x_C) f(x_C) \, \mathrm{d} \mu_C(x_C)?\)
2. Is the causal effect of \(X_T\) on \(X_R\) identifiable? That is, is \(P(X_R \in \cdot \, ; \doOp(X_T = x_T^*))\) uniquely determined by the marginal distribution of \((X_T, X_R, X_C)\)?