SCM (Structural Causal Model)

Definition

An SCM (Structural Causal Model) is a framework for mathematically expressing the causal relationships among variables. It is the core of Pearl’s causal inference framework.

Formal Definition:

$M = (V, U, F, P)$

Component	Meaning
$V$	Endogenous variables (observable variables)
$U$	Exogenous variables (unobserved, $V \cap U = \emptyset$)
$F$	Structural equations: $f_i: (V \cup U)^p \rightarrow V$
$P$	Distribution over exogenous: $P(U) = \prod_i P(U_i)$

Structural Equation: $X_i := f_i(Pa(X_i), U_i)$

Each variable is determined by its parent variables and exogenous noise.

Example

Causal Graph G:           SCM M:
    X                     V = {X, Y, Z}
    ↓                     U = {U_X, U_Y, U_Z}
    Y
    ↓                     F = { X := U_X,
    Z                           Y := f(X) + U_Y,
                                Z := g(Y) + U_Z }

                          P = { U_X ~ N(0,1),
                                U_Y ~ N(0,1),
                                U_Z ~ N(0,1) }

Joint Distribution (Markov factorization): $P(X, Y, Z) = P(X) \cdot P(Y|X) \cdot P(Z|Y)$

SCM vs DAG

Aspect	DAG	SCM
Representation	Qualitative (presence of arrows)	Quantitative (functional form)
Information content	Causal direction only	Causal direction + functional relationships
Use	Identification	Identification + Estimation
Intervention	Conceptual representation	Mathematically manipulable

Causal Edge Assumption

The value of each variable is fully determined by its parent variables: $X_i := f(Pa(X_i), U_i) \quad \forall X_i \in V$

Meaning:

• Causal sufficiency: all common causes are observed
• No unmeasured confounders (in ideal case)

Intervention (do-operator)

Definition

$do(X = x)$: forcibly set the variable $X$ to the value $x$

Mathematical manipulation:

1. Replace the structural equation of $X$ with $X := x$
2. Keep the other equations

Example

Original SCM:

X := U_X
Y := 2X + U_Y
Z := Y + U_Z

After $do(X = 3)$:

X := 3          ← Changed!
Y := 2X + U_Y   ← Uses X = 3
Z := Y + U_Z

Interventional vs Observational

$P(Y | X = x) \neq P(Y | do(X = x))$

• Observational: observational conditional (includes confounding)
• Interventional: causal intervention (removes confounding)

Causal Effect Identification

From SCM to DAG

Extract a DAG $G$ from an SCM $M$:

• Each variable in $V$ is a node
• $Pa(X_i) \rightarrow X_i$ edge

do-calculus

Pearl’s three rules convert an interventional distribution into an observational one:

Rule 1 (Insertion/deletion of observations): $P(y|do(x), z, w) = P(y|do(x), w)$ if $(Y \perp Z | X, W)_{G_{\overline{X}}}$

Rule 2 (Action/observation exchange): $P(y|do(x), do(z), w) = P(y|do(x), z, w)$ if $(Y \perp Z | X, W)_{G_{\overline{X}\underline{Z}}}$

Rule 3 (Insertion/deletion of actions): $P(y|do(x), do(z), w) = P(y|do(x), w)$ if $(Y \perp Z | X, W)_{G_{\overline{X}\overline{Z(W)}}}$

Counterfactuals

An SCM enables counterfactual reasoning:

Counterfactual query: “If X had been x’, what would Y have been?”

$Y_{X=x'} = Y(M_{do(X=x')})$

Three-step procedure:

1. Abduction: Observe evidence, infer $U$
2. Action: Modify SCM with $do(X=x')$
3. Prediction: Compute $Y$ in modified model

Linear SCM

Linear Gaussian SCM: $X_i = \sum_{j \in Pa(i)} \beta_{ji} X_j + U_i, \quad U_i \sim N(0, \sigma_i^2)$

Matrix form: $\mathbf{X} = B\mathbf{X} + \mathbf{U}$ $\mathbf{X} = (I - B)^{-1}\mathbf{U}$

Characteristics:

• A closed-form solution exists
• LiNGAM: identifiable with non-Gaussian noise

Causal Discovery from SCM

Goal

Recover the underlying SCM (or DAG) from data $D$

Approaches

1. Constraint-based (PC, FCI): Conditional independence tests
2. Score-based (GES, FGES): Score optimization
3. Asymmetry-based (LiNGAM): Distributional asymmetries

Identifiability

• Markov Equivalence: same conditional independencies → same Markov Equivalence Class
• Non-Gaussian: a unique DAG can be identified with LiNGAM

Related Concepts

• DAG - Graphical representation of an SCM
• Confounder - Unmeasured common cause
• d-separation - Graphical conditional independence
• Markov Equivalence Class - Observationally equivalent graphs
• Back-door Criterion - Causal effect identification
• Potential Outcomes - Alternative causal framework
• Markov Property - Graph-distribution relationship
• Graph Foundations Overview - Complete overview of graphical representations

References

• Pearl, J. (2009). Causality: Models, Reasoning, and Inference
• yaoSurveyCausalInference2021 - SCM in causal discovery context