[KSSCI 2021] 인과추론의 데이터 과학 - Session 7/8 보충

인과추론의 데이터과학. (2021, Oct 1). [Session 7/8 - 보충 1] 베이지안 네트워크 (Bayesian Network)
[Video]. YouTube.
인과추론의 데이터과학. (2021, Oct 1). [Session 7/8 - 보충 2] 베이지안 네트워크에서의 상관관계 증명
[Video]. YouTube.

 

Session 7/8 보충 1

Probability

• \(P(A)\) : unconditional / marginal Pr
  • marginal probability: 주변 확률 (=개별 사건의 확률)
• \(P(A|B)\) : conditional Pr
• joint Pr
  $$ P(B|A) = \displaystyle{{P(B)P(A|B)}\over P(A)} $$
  $$ \begin{align*} P(A\cap B) &= P(A,B) \\ &= P(A)P(B|A)\\ &= P(B)P(A|B)\end{align*} $$
• \(P(A, B, C)\)
  $$ \begin{align*} P(A,B,C) &= P(A)P(B,C|A) \\ &= P(A)P(B|A)P(C|A,B)\end{align*} $$

Marginalize

• Conditional Pr / Joint Pr ⇒ Marginal Pr
• \(P(A=0) = P(B=0)P(A=0|B=0) + P(B=1)P(A=0|B=1)\)
  
  ⇒ \(P(A) = \displaystyle\sum_B{P(B)P(A|B)} = \sum_B{P(A,B)}\)

Independent

• dependent : correlation, association이 있는 것
  • \(P(A, B) \ne P(A)P(B)\)
• independent: \(A\perp B\)
  • \(P(A|B) = P(A)\)
  • \(P(A,B) = P(A)P(B|A) = P(A)P(B)\)

• conditional independent
  • \(P(A,B|C) = P(A|C)P(B|C)\) ⇒ \(C\)라는 조건 하에, \(A\)와 \(B\)가 independent하다

Causal Markov Assumption

• \(P(X,Y,Z) = P(X)P(Y,Z|X) = P(X)P(Y|X)P(Z|X,Y)\)
• Under DAG(Causal Graph) + Markov Assumption ⇒ Bayesian Network Factorization
  • \(X → Y → Z​\)
  • \(P(X,Y,Z) = P(X)P(Y|X)P(Z|Y)\)

 

Session 7/8 보충 2

Mediator

• \(X → M → Y\)
• Bayesian Network Factorization: \(P(X, Y, M) = P(X)P(M|X)P(Y|M)\)
• Objective: \(P(X, Y) = P(X)P(Y)\) ??
  • if \(P(X, Y) = P(X)P(Y)\) ⇒ independent
  • if \(P(X, Y) \ne P(X)P(Y)\) ⇒ dependent
• Proof
  • \(P(M|X) = P(M)\) ?? ⇒ \(P(M|X) \ne P(M)\)
    • if \(P(M|X) = P(M)\)
    ⇒ \(P(X)\displaystyle\sum_M{P(M)P(Y|M)} = P(X)\sum_M{P(Y, M)} = P(X)P(Y)\)
    • \(M\perp X\) (DAG)

$$ \begin{align*}P(X,Y) = \displaystyle\sum_M{P(X,Y,M)} &= \sum_M{P(X)P(M|X)P(Y|M)}\\&=P(X)\sum_M{P(M|X)P(Y|M)}\\&\ne P(X)P(Y)\end{align*} $$

• Conditioning: \(X → M(Condition) → Y\)
  • Objective: \(P(X,Y|M) = P(X|M)P(Y|M)\) ??
  • Bayesian Network Factorization: \(P(X,Y,M) = P(X)P(M|X)P(Y|M)\)
  • Proof
    • \(P(X,Y,M) = P(M)P(X,Y|M)\)
    $$ \begin{align*}P(X,Y) &= P(X,Y|M) \\&= \displaystyle{P(X)P(M|X)P(Y|M)\over P(M)}\\ &= {P(X)P(M|X)\over P(M)}P(Y|M) \\&= P(X|M)P(Y|M)\end{align*} $$

Confounder

• \(X ← C → Y\)
• Objective: \(P(X,Y) = P(X)P(Y)\) ??
• Bayesian Network Factorization: \(P(X,Y,C) = P(X|C)P(Y|C)P(C)\)
• Proof
  • \(P(X)P(Y) = P(Y)\displaystyle\sum_C{P(X,C)} = P(Y)\sum_C{P(C)P(X|C)}\)
  $$ \begin{align*}P(X,Y) &= \displaystyle\sum_C{P(X,Y,C)} \\&= \sum_C{P(X|C)P(Y|C)P(C)}\\ &=\sum_C{P(X|C)P(Y)P(C|Y)}\\&=P(Y)\sum_C{P(X|C)P(C|Y)}\end{align*} $$
  • \(\displaystyle P(Y)\sum_C{P(C)P(X|C)} = P(Y)\sum_C{P(X|C)P(C|Y)}\) ??
    
    ⇒ \(P(C) = P(C|Y)\) ?? ⇒ dependent ⇒ \(P(C) \ne P(C|Y)\)
    
    ⇒ \(P(X,Y) \ne P(X)P(Y)\)
• Conditioning: \(X ← C(Condition) → Y\)
  • Objective: \(P(X,Y|C) = P(X|C)P(Y|C)\) ??
  • Bayesian Network Factorization: \(P(X,Y,C) = P(X|C)P(Y|C)P(C)\)
  • Proof
    • \(P(X,Y,C) = P(C)P(X,Y|C)\)
    • \(P(X,Y|C) = P(X|C)P(Y|C)\) ⇒ conditional independent

Collider

• \(X → C ← Y\)
• Objective: \(P(X,Y) = P(X)P(Y)\) ??
• Bayesian Network Factorization: \(P(X,Y,C) = P(X)P(Y)P(C|X,Y)\)
• Proof
  • \(\displaystyle\sum_C{P(C|X,Y)} = 1\)
  $$ \begin{align*}P(X,Y) &= \displaystyle\sum_C{P(X,Y,C)} \\&= \sum_C{P(X)P(Y)P(C|X,Y)}\\ &=P(X)P(Y)\sum_C{P(C|X,Y)}\\&=P(X)P(Y)\end{align*} $$
• Conditioning: \(X → C(Condition) ← Y\)
  • Objective:
    • \(P(X,Y|C) = P(X|C)P(Y|C)\) ??
    • ✅ \(P(X|C) = P(X|C,Y)\) ??
  • Bayesian Network Factorization: \(P(X,Y,C) = P(X)P(Y)P(C|X,Y)\)
  • Proof
    • \(P(X|C) = \displaystyle {P(X)P(C|X)\over P(C)}\)
    • \(P(X,C|Y) = P(X|Y)P(C|Y, X) = P(C|Y)P(X|Y,C)\)
      • \(P(X|Y,C) = \displaystyle {P(X|Y)P(C|Y,X)\over P(C|Y)}\)
    • \(X\), \(Y\)는 dependent → \(P(X) = P(X|Y)\)
      • \(\displaystyle{P(C|X)\over P(C)} = {P(C|Y,X)\over P(C|Y)}\) ??
      • 좌변: Effect of \(X\) on \(C\)
      • 우변: Effect of \(X\) on \(C\) after controlling for \(Y\)
      ⇒ 어떤 통제변수를 넣으면, 다른 변수도 영향을 받을 수 밖에 없음 (ex. multivariate regression)
      
      ⇒ \(\displaystyle{P(C|X)\over P(C)} \ne {P(C|Y,X)\over P(C|Y)}\)

Correlation ≠ Causation

• Confounder : \(X ← C → Y ← X\)
• Bayesian Network Factorization: \(P(X,Y,C) = P(X|C)P(Y|C,X)P(C)\)
• Proof
  • \(P(X)P(Y,C|X) = P(X|C)P(Y|X,C)P(C)\)
  • \(P(Y,C|X) = \displaystyle{P(X|C)P(C)P(Y|X,C)\over P(X)} = P(C|X)P(Y|X,C)\)
    • \(\displaystyle{P(X|C)P(C)\over P(X)} = {P(X,C)\over P(X)} = {P(X)P(C|X)\over P(X)} = P(C|X)\)
  • \(P(Y|X) = \displaystyle\sum_C{P(Y,C|X)} = \sum_C{P(C|X)P(Y|X,C)}\)
• do operator : \(C → Y ← do(X)\)
  • causal effect: \(P(Y|do(X))\)
  • Bayesian Network Factorization: \(P(do(X), Y, C) = P(do(X))P(Y|do(X), C)P(C)\)
  • Proof
    • \(P(do(X), Y, C) = P(do(X))P(Y,C|do(X))\)
    • \(P(Y|do(X)) = \displaystyle\sum_C{P(Y,C|do(X))} = \sum_C{P(Y|do(X),C)P(C)}\)
• \(\displaystyle\sum_C{P(C|X)P(Y|X,C)} = \sum_C{P(Y|do(X),C)P(C)}\) ??
  • \(\displaystyle\sum_C{P(C|X)} = \sum_C{P(C)}\) ??
    • \(do(X) → X\)
  • \(X → C\) ⇒ dependent ⇒ \(P(C|X) \ne P(C)\) ⇒ correlation ≠ causation