Conditional independence of two random variables given a third random variable

Question

I have three random variables $(X, Y, Z)$. The variables $X$ and $Y$ are conditionally independent given $Z$.

Will it be correct if I assume $P(X | Z) = P(X)$ or $P(X | Y) = P(X)$ ? If not, then why? Can you give a simple example to explain why the above assumption will not be correct?

Answer 1

Let $X:=\epsilon_1+Z$ and $Y:=\epsilon_2+Z$, where $\epsilon_1$ and $\epsilon_2$ are independent $N(0,1)$ (independent of $Z$) and $\mathsf{P}(Z=1)=\mathsf{P}(Z=-1)=1/2$. $$ \mathsf{P}(X\le 0\mid Z)=\mathsf{P}(\epsilon_1+Z\le 0\mid Z)=\Phi(-Z). $$ But $\mathsf{P}(X\le 0)=1/2$.

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Conditional independence means that for all Borel sets $A$ and $B$, $$ \mathsf{P}(X\in A, Y\in B\mid Z)=\mathsf{P}(X\in A\mid Z)\mathsf{P}(Y\in B\mid Z). $$