Mutual information and chain rule

Question

I'm having difficulty proving the below statement. It looks trivial but I don't know how to treat the term $XY$.

$I(XY ;Z) = I(X;Z)+ I(Y ;Z|X)$

How is it different form $I(X,Y ;Z) = I(X;Z)+ I(Y ;Z|X)$?

Answer 1

You can write: \begin{align*} I(X,Y;Z)&=H(X,Y)-H(X,Y|Z)\\ &=H(X)+H(Y|X)-(H(X|Z)+H(Y|X,Z))\\ &=(H(X)-H(X|Z))+(H(Y|X)-H(Y|X,Z))\\ &=I(X;Z)-I(Y;Z|X) \end{align*}