Confusion about conditional entropy when conditioning on another event

Question

Suppose $X$ and $Y$ are discrete random variables. I would like to relate the entropy $H[X \mid Y]$ to the same entropy conditioned on the additional event $y>0$.

My reasoning is as follows: \begin{align} H[ X \mid Y] &= \sum_y p(y) H[ X \mid Y=y] \\ &= \sum_{y >0} p(y) H[ X \mid Y=y, y>0] \\ &+ \sum_{y \le 0} p(y) H[ X \mid Y=y, y\le 0] \end{align} So if I know $H[ X \mid Y=y, y>0]$, the above reasoning would allow me to obtain a lower bound for $H[ X \mid Y]$ by using $H[ X \mid Y=y, y>0]$ assuming I know something about $p(y>0)$.

Is my line of thought correct?

If yes, then I'm genuinely puzzled because of the following:

Let $I$ be an indicator random variable that is $1$ iff $y>0$. Now it appears to me that the right-hand side above is equivalent to $H[ X \mid Y, I]$. Therefore wouldn't the above imply that $H[ X \mid Y] = H[ X \mid Y, I]$ (which isn't true in general)?

Answer 1

Your conclusion that $H(X|Y)=H(X|YI)$ in this case is in fact correct, because the indicator variable $I$ you have constructed is a function of $Y$ alone. In such situations the "data-processing inequality" $H(X|Y)\geq H(X|YI)$ for conditional entropy is saturated (one way to look at it is that you can simply apply the data-processing argument in the reverse direction as well, because the pair of random variables $YI$ can be generated from $Y$ alone).

Another perspective on this is that in general we have a chain rule $H(X|YI)=H(X|Y)-\mathcal{I}(I:X|Y)$ (using $\mathcal{I}$ for mutual info to avoid notation clash with the indicator variable), but since in your case $I$ is a function of $Y$ alone, the $\mathcal{I}(I:X|Y)$ term is equal to zero (e.g. by noting $X \leftrightarrow Y \leftrightarrow I$ form a Markov chain).