# Simple expression related to Mutual Information

One way to define the mutual information is

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

I have found it useful to look the related quantity

$$?(X;Y=y) = H(X) - H(X|Y=y)$$

That is, we look at how much the entropy of $$X$$ is decreased given a particular outcome $$y$$ for $$Y$$.

It is not hard to see that the mutual information is regained on expectation on $$Y$$, so that we have

$$E_Y[?(X;Y=y)] \\ = \sum_y p(y)(H(X) - H(X|Y=y)) \\ = H(X) - \sum_y p(y)H(X|Y=y) \\ = H(X) - H(X|Y) \\ = I(X;Y)$$

My question: does my $$?(X;Y=y)$$ function have a name? Or a standardized notation?

Note it's not the same as pointwise mutual information: rather, I think this would be the expectation of pointwise mutual information, but only on $$X$$ (rather than both variables). So it's in between the regular mutual information and the pointwise version.