I have a (perhaps simple) question about the chain rule for mutual information. The formula is given by
$$I(X_1, X_2, ..., X_n; Y) = \sum_{i=1}^{n} I(X_i; Y| X_{i-1}, X_{i-2}, ..., X_1)$$
My question is how to use it on the following equation:
$$I(X; Y_1, Y_2) = I(X; Y_1) + I(X; Y_2| Y_1) \hspace{3cm} (1)$$
I just get $I(X; Y_1, Y_2) = I(Y_1, Y_2; X) = I(X; Y_1) + I(Y_2; X|Y_1)$
With the definition of conditional information and the chain rule for entropy it follows that
$I(X,Y_2|Y_1) = H(X|Y_1) - H(Y_2,X|Y_1) + H(Y_2|Y_1)$
and
$I(Y_2;X|Y_1) = H(Y_2|Y_1) - H(X,Y_2|Y_1) + H(X|Y_1)$
This leads to the equation $H(Y_2, X|Y_1) = H(X, Y_2| Y_1)$, which is obviously not true. Where do I make a mistake? Also, how can i derive euqation (1) directly by the chain rule? I also followed the proof on wikipedia, which I understood. But I still don't see it from the chain rule for mutual information.
Thank you very much for your help and patience :)