Consider five random variables V, U, W, X, Y whose joint distribution satisfies: P(V, U, W, X, Y) = P(V) * P(U) * P(W | U, V) * P(X | W) * P(Y | W) Which ONE of the following statements is FALSE? A. Y is conditionally independent of V given W B. X is conditionally independent of U given W C. U and V are conditionally independent given W D. Y and X are conditionally independent given W

GATE 2024 · Engineering Mathematics · Bayesian Network · medium

Answer: C. U and V are conditionally independent given W — this is FALSE because W is a collider, and conditioning on a collider opens the path between its parents U and V.

1. Identify the network structure: Edges: V->W, U->W, W->X, W->Y. V and U have no parents. W is a collider for U and V.
2. Check option A: Y conditionally independent of V given W: The only path from Y to V passes through W. W is a non-collider on this path (since V->W->Y). Conditioning on W blocks this path. So Y _|_ V | W is TRUE.
3. Check option B: X conditionally independent of U given W: The path from X to U goes X<-W<-U. Conditioning on W (chain node / non-collider) blocks this path. So X _|_ U | W is TRUE.
4. Check option C: U and V conditionally independent given W: The path U->W<-V has W as a collider. Conditioning on a collider activates (opens) the path. Therefore U and V are NOT conditionally independent given W. This statement is FALSE.
5. Check option D: Y and X conditionally independent given W: The path from Y to X is Y<-W->X. W is a fork node (non-collider). Conditioning on W blocks this path. So Y _|_ X | W is TRUE.