GATE DA 2024 | Question 24

MCQ · 1 mark · −0.33

Consider five random variables \(U, V, W, X\), and \(Y\) whose joint distribution satisfies:

\[P(U, V, W, X, Y) = P(U)P(V)P(W \mid U, V)P(X \mid W)P(Y \mid W)\]

Which ONE of the following statements is FALSE?

  1. Option A: \(Y\) is conditionally independent of \(V\) given \(W\)
  2. Option B: \(X\) is conditionally independent of \(U\) given \(W\)
  3. Option C: \(U\) and \(V\) are conditionally independent given \(W\) (correct)
  4. Option D: \(Y\) and \(X\) are conditionally independent given \(W\)

Answer

(C)

Explanation

The factorisation is the Bayesian network \(U \rightarrow W \leftarrow V\), \(W \rightarrow X\), \(W \rightarrow Y\).

  • A and B: \(U\) and \(V\) reach \(X\) and \(Y\) only through chains such as \(V \rightarrow W \rightarrow Y\), and observing \(W\) blocks a chain. Both are true.
  • D: \(X \leftarrow W \rightarrow Y\) has \(W\) as a common cause, so \(X\) and \(Y\) are independent once \(W\) is known. True.
  • C: \(U\) and \(V\) are independent on their own (the factor \(P(U)P(V)\)), but \(W\) is their common effect. Observing a common effect couples its causes (explaining away): once \(W\) is known, learning \(U\) changes what \(V\) is likely to be. So C does not hold in general.

Answer: C