Let $W,X,Y,Z$ be random variables. Then $ X\perp{Z,W} | Y $ implies $X\perp{Z|Y}$.
can you please help me prove it? I'm sure it's easy but got stuck for some reason.
Intuitively the lemma is clear. if given Y, X and W,Z are conditionally independent, then it doesn't matter what values W and Z will take it won't effect X. So, it means that it doesn't matter what values Z will take it won't effect X. So X and Z are CI given Y. I guess that I can try proof by contradiction but I am pretty sure it is not elegant nor necessary. I tried to include W somehow to use what we know but couldn't figure out how to use it in an effective manner.
Well Thought about it some more and got the answer:
$ P[X,Z|Y] = \sum_{w}P[X,Z,W|Y] =^{assumption} \sum_{w}P[X|Y] \cdot P[Z,W|Y] = P[X|Y] \cdot \sum_{w} P[Z,W|Y] = P[X|Y] \cdot P[Z|Y] $