Consider five random variables π, π, π, π, and π whose joint distributionβ¦
2024
Consider five random variables π, π, π, π, and π whose joint distribution satisfies:
π(π, π, π, π, π) = π(π)π(π)π(π|π, π)π(π|π)π(π|π)
Which ONE of the following statements is FALSE?
- A.
π is conditionally independent of π given π
- B.
π is conditionally independent of π given π
- C.
π and π are conditionally independent given π
- D.
π and π are conditionally independent given π
Attempted by 4 students.
Show answer & explanation
Correct answer: C
Answer: The false statement is that U and V are conditionally independent given W.
Reasoning:
Start from the given factorization: P(U,V,W,X,Y)=P(U)P(V)P(W|U,V)P(X|W)P(Y|W).
Y is conditionally independent of V given W because the factor for Y is P(Y|W); thus P(Y|W,U,V,X)=P(Y|W).
X is conditionally independent of U given W because the factor for X is P(X|W); thus P(X|W,U,V,Y)=P(X|W).
X and Y are conditionally independent given W because the joint conditional factorizes as P(X,Y|W)=P(X|W)P(Y|W).
U and V are not conditionally independent given W. W is a common effect (a collider) of U and V: U β W β V. Conditioning on a collider generally creates dependence (explaining-away). Formally,
P(U,V|W) β P(U)P(V)P(W|U,V),
which does not in general factor into a product of a function of U and a function of V alone, so U and V are generally dependent once W is observed.
Conclusion: The only false statement is the claim that U and V are conditionally independent given W; the other three statements follow directly from the given factorization.