Understanding problem of chain Rule for mutial information

Question

So I have to proof $ I(X;Z|Y) = I(Z;Y|X) - I(Z;Y) + I(X;Z) $

Write mutual information in terms of entropy or use the chain rule for mutual information for an immediate proof.

So I want to directly proof using chain rule:

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

$ I(X;Z|Y) = I(Y;X;Z) - I(Y;Z) $

Now I got half of the proof, now I need to get the other two parts from the chain rule, however as I undersand the chain rule I cannot get $ I(X;Z) + I(Z;Y|X) $ from the chain rule directly?

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

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

Is there another possibility using chain rule I do not know?

Answer 1

I believe you expanded the chain rule the wrong way. You should have

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

and also

$$I(X,Y;Z) = I(Y;Z) + I(Z;X|Y).$$

Thus

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

and re-arranging gives the result.