Conditional independence: does $(X \bot Y \mid Z) \land (X \bot Y \mid W) \implies (X \bot Y \mid Z , W)$?

Question

I'm reading a book about probabilistic graphical models by Daphie Koller and Nir Friedman and I'm stuck at the following exercise:

Is it true that $ (X \bot Y \mid Z) \land (X \bot Y \mid W) \implies (X \bot Y \mid Z , W) $?

Any ideas how to prove or disprove the statement? $X \bot Y \mid Z$ denotes conditional independence of X and Y given Z, i.e., $P(X\mid Z)=P(X\mid Y,Z)$.

Answer 1

It's not true. For a counterexample, let $X$, $Y$, and $Z$ be i.i.d., each with equal probabilities of being $+1$ and $-1$, and let $W=XYZ$. Any three of $X$, $Y$, $Z$ and $W$ are i.i.d., but once $Z$ and $W$ are known $X$ and $Y$ are not independent as $XY=ZW$.