Solution

The question requires identifying the incorrect statement about conditional independence among five random variables based on their joint probability distribution.

Dependency Structure Analysis

The joint probability distribution is provided as:

$ P(U,V,W,X,Y) = P(U)P(V)P(W|U,V)P(X|W)P(Y|W) $

This factorization reveals the underlying dependencies:

• $W$ is directly dependent on $U$ and $V$.
• $X$ depends solely on $W$.
• $Y$ depends solely on $W$.

We can represent this as a Bayesian Network graph: $U \rightarrow W \leftarrow V$ $W \rightarrow X$ $W \rightarrow Y$ In this structure, $W$ acts as a common effect for $U$ and $V$, and a common cause for $X$ and $Y$. $U$ and $V$ are initially independent. $X$ and $Y$ are independent conditional on $W$.

Evaluating Conditional Independence Statements

Conditional independence $A \perp B | C$ holds if $P(A|B,C) = P(A|C)$. We examine each option:

1. $Y$ is conditionally independent of $V$ given $W$ ($Y \perp V | W$)

The dependency path between $Y$ and $V$ is through $W$: $Y \leftarrow W \rightarrow V$. Since $W$ is a common ancestor (or cause), conditioning on $W$ blocks this path. Therefore, $Y$ and $V$ are conditionally independent given $W$. This statement is TRUE.

2. $X$ is conditionally independent of $U$ given $W$ ($X \perp U | W$)

The path is $U \rightarrow W \rightarrow X$. $W$ is an intermediate variable. Conditioning on $W$ breaks the dependency flow from $U$ to $X$. The term $P(X|W)$ in the factorization confirms that $X$'s probability depends only on $W$, not $U$, once $W$ is known. This statement is TRUE.

3. $U$ and $V$ are conditionally independent given $W$ ($U \perp V | W$)

The term $P(W|U,V)$ explicitly states that $W$ depends on both $U$ and $V$. In the graph $U \rightarrow W \leftarrow V$, $W$ is a common effect. If $W$ is observed, information about $W$ influences our beliefs about its causes, $U$ and $V$. They become dependent (explaining away effect). Thus, $U$ and $V$ are NOT conditionally independent given $W$. This statement is FALSE.

4. $Y$ and $X$ are conditionally independent given $W$ ($Y \perp X | W$)

The path is $X \leftarrow W \rightarrow Y$. $W$ is a common parent (cause). Conditioning on $W$ blocks this path. The factorization includes separate terms $P(X|W)$ and $P(Y|W)$, directly indicating that $X$ and $Y$ are conditionally independent given $W$. This statement is TRUE.

Conclusion

The statement "$U$ and $V$ are conditionally independent given $W$" is the only false statement.