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!