I am looking for a proof for a formula seen on Wikipedia:

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

I checked the ref given on Wiki but couldn't find the formula in it.

Thank you!

  • 2
    Note that $H(Z|X) - H(Z) = -I(X;Z)$, and $H(Z|Y) - H(Z|X,Y) = I(Z;X|Y)$. Consequently, the above relation is $I(X;Y|Z) \overset{?}= I(X;Y) + I(X;Z|Y) - I(X;Z)$, which is easy to show by expanding $I(X;Y,Z)$ by the chain rule in two different ways. – stochasticboy321 Jul 19 '16 at 7:23
  • Thank you for teaching me humility :-). – Robinson Jul 20 '16 at 0:10

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