I am trying to solve this problem:
We have three random variables: $X,Y, Z =f(X)$ I want to calculate $\frac{\partial I(Z;Y)}{\partial p(z_0|x_0)}$ In particular, since $p(z)=\sum_x p(z|x)p(x)$ i could re-write $I(Z;Y)=\sum_{z,y} p(y|x)(\sum_x p(z|x)p(x)) \log{\frac{p(y|z)}{p(y)}}$ erasing the dependence from $p(z)$ inside the logarithm, but is it correct? Is there still a dependence due to $p(y|z)$? In the end I came up with something like $\sum_{z,y}p(y|z)p(x_0)\log{\frac{p(y|z)}{p(y)}}$ that doesn't seem to be very interesting, is it?
Thanks for any kind of help