Interventional Distributions

A family of interventional distributions is a collection
\(\bigl\{\,P_{A,x_A}\colon A\subseteq V,\ x_A\in\mathbb R^A\bigr\},\) where each \(P_{A,x_A}\) is a joint law for \(X=(X_v)_{v\in V}\) such that under \(P_{A,x_A}\),
\(X_A = x_A\quad\text{with probability }1.\) Notation:
\(P\bigl(X\in\cdot\,;\,\doOp(X_A=x_A)\bigr)=P_{A,x_A}.\)

Causal Markov Property

Definition
A family of interventional distributions satisfies the causal Markov property for the DAG \(G=(V,E)\) if for every \(A\subseteq V\) and \(x_A\in\mathbb R^A\):

1. \(P\bigl(X\in\cdot\,;\,\doOp(X_A=x_A)\bigr)\) factorizes according to \(G\).
2. For all \(v\notin A\), the interventional conditional law equals the observational one: \(P\bigl(X_v\in\cdot\mid X_{\pa(v)}=x_{\pa(v)};\doOp(X_A=x_A)\bigr) = P\bigl(X_v\in\cdot\mid X_{\pa(v)}=x_{\pa(v)}\bigr),\) whenever the conditioning values agree with the intervention.

Truncated Factorization

Let \(f\) be the observational joint density (w.r.t.\ \(\mu=\bigotimes_{v}\mu_v\)), and write
\(f(x_v\mid x_{\pa(v)})\) for the conditional densities.

Proposition
A family of interventional distributions satisfies the causal Markov property if and only if
\(f\bigl(x;\,\doOp(x_A=x_A)\bigr) = \prod_{v\notin A} f(x_v\mid x_{\pa(v)}) \;\times\; \prod_{v\in A} \mathbf{1}\{x_v = x_A\},\) with the dominating measure \(\mu_{V\setminus A}\times\) counting on \(\mathbb R^A\).

Mutilated DAG

Definition
For a DAG \(G=(V,E)\) and \(A\subseteq V\), the mutilated DAG \(G_{\doOp(A)}=(V,E_{\doOp(A)})\) has
\(E_{\doOp(A)} \;=\; E\setminus\{\,w\to v: v\in A\}.\)

Proposition
Under the intervention \(\doOp(X_A=x_A)\), the interventional distribution factorizes according to the mutilated DAG: \(P\bigl(X\in\cdot\,;\,\doOp(X_A=x_A)\bigr) \quad\text{factorizes on }G_{\doOp(A)}.\)

Structural Equation Models

A structural equation model (SEM) on \(G=(V,E)\) assumes \(X_v \;=\; g_v\bigl(X_{\pa(v)},\,\omega_v\bigr),\quad v=1,\dots,m,\) where the “noise” variables \(\omega_1,\dots,\omega_m\) are random.

Proposition
If \(\omega_1,\dots,\omega_m\) are independent, then the joint law of the unique solution \(X\) satisfies the local Markov property for \(G\).

Fact
Conversely, every distribution \(P_X\) satisfying the local Markov property for \(G\) arises from some SEM with independent errors (e.g.\ one may take \(\omega_v\sim\mathrm{Unif}(0,1)\) and choose \(g_v\) as the quantile map of \(P(X_v\mid X_{\pa(v)})\)).

Structural Causal Models

Interpreting SEMs causally, one writes assignments
\(X_v := g_v\bigl(X_{\pa(v)},\omega_v\bigr)\) and under \(\doOp(X_A=x_A)\) replaces each equation for \(v\in A\) by
\(X_v := x_A.\) The resulting variables \(X(\doOp(X_A=x_A))\) are called potential outcomes.