# Some property of conditional independence

Given random variables $Y, Z, X_1, X_2$

Is there some relation between

• $Y$ and $Z$ are conditionally independent given $(X_1, X_2)$
• $Y$ and $Z$ are conditionally independent given $X_1$, and given $X_2$

My guess is that the former implies the latter, but not vice versa, since the sigma algebras generated by the random variables $\sigma(X_1),\sigma(X_2) \subseteq \sigma(X_1, X_2)$?

When will they become equivalent?

Is there some reference which state properties of conditional independence? Thanks!

Let $$Y$$, $$X_1$$ and $$X_2$$ be independent random variables with uniform distribution on $$[0,1]$$.
Now let $$Z=X_1+X_2+Y\mod 1.$$
Then the random variables $$Z$$ and $$Y$$ are independent conditionally on $$X_1$$ and conditionally on $$X_2$$ but clearly not conditional on $$X_1$$ and $$X_2$$. So the second condition does not imply the first one.
The first condition does not imply the second one. Take a constant random variable $$X_1$$. Then the conditional independence between $$Y$$ and $$Z$$ given $$(X_1,X_2)$$ is the conditional independence between $$Y$$ and $$Z$$ given $$X_2$$, and the conditional independence between $$Y$$ and $$Z$$ given $$X_1$$ is the independence between $$Y$$ and $$Z$$. It is possible that $$Y$$ and $$Z$$ are conditionally independent given $$X_2$$, but not independent.