SCM (Structural Causal Model)

Definition

SCM (Structural Causal Model)은 변수들 간의 인과 관계를 수학적으로 표현하는 framework. Pearl의 causal inference framework의 핵심.

Formal Definition:

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

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

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

각 변수는 부모 변수들과 외생적 noise에 의해 결정됨.

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
표현	Qualitative (화살표 유무)	Quantitative (함수 형태)
정보량	인과 방향만	인과 방향 + 함수 관계
용도	Identification	Identification + Estimation
Intervention	개념적 표현	수학적 조작 가능

Causal Edge Assumption

각 변수의 값은 부모 변수들에 의해 완전히 결정: $X_i := f(Pa(X_i), U_i) \quad \forall X_i \in V$

의미:

• Causal sufficiency: 모든 공통 원인이 관측됨
• No unmeasured confounders (in ideal case)

Intervention (do-operator)

Definition

$do(X = x)$: 변수 $X$를 값 $x$로 강제 설정

수학적 조작:

1. $X$의 structural equation을 $X := x$로 대체
2. 다른 equations는 유지

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: 관측 조건부 (confounding 포함)
• Interventional: 인과적 개입 (confounding 제거)

Causal Effect Identification

From SCM to DAG

SCM $M$에서 DAG $G$ 추출:

• $V$의 각 변수가 node
• $Pa(X_i) \rightarrow X_i$ edge

do-calculus

Pearl의 세 가지 규칙으로 interventional distribution을 observational로 변환:

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

SCM은 counterfactual reasoning 가능:

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

특징:

• Closed-form solution 존재
• LiNGAM: Non-Gaussian noise로 identifiable

Causal Discovery from SCM

Goal

데이터 $D$로부터 underlying SCM (또는 DAG)을 복원

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: 같은 conditional independencies → 같은 Markov Equivalence Class
• Non-Gaussian: LiNGAM으로 unique DAG 식별 가능

Related Concepts

• DAG - SCM의 graphical representation
• 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 - 그래프-분포 관계
• Graph Foundations Overview - 그래프 표현 전체 정리

References

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