GATE DS&AI 2024 | Question: 14

Question

​​​​​​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?

• 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 $W$

Answer 1

$\begin{align} P(U, V, W, X, Y) &= P(U)P(V|U)P(W|U, V)P(X|W, U, V)P(Y|W, U, V, X) \\ &= P(V)P(U|V)P(W|U, V)P(X| W, U, V)P(Y| W, U, V, X) \\ &= P (V)P(U|V)P(W|U, V)P(Y| W, U, V)P(X| W, U, V, Y) \\ &= P(U)P(V| U)P(W|U, V)P(Y| W, U, V)P(X| W, U, V, Y) \\ &= P(U)P(V)P(W|U, V)P(X| W)P(Y| W) \end{align}$

From above we can see,

A. $P(Y|W, U, V) = P(Y| W) \implies$ Y is conditionally independent of V given W. Option A can beTrue.

B. $P(X|W, U, V) = P(X| W) \implies$ X is conditionally independent of U given W. Option B can be True.

C.$P(U|V) = P(U), P(V|U) = P(V) \implies$ U and V are independent. But from this we can't say whether option C is either True or False.

D. From A and B, we can say this can be also True.

Answer - C

Answer 2

The Bayes Net for the given expression is shown below:

Note that there are no arrows to U and V. Hence, we can say U and V are conditionally independent, given W, X and/or Y.

$\therefore$ option C.