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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)?

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  • $\begingroup$ how did you get that the above is equal to H[X∣Y,I] ? Maybe i missed something obvious? $\endgroup$ Feb 17, 2020 at 13:47
  • $\begingroup$ besides for that, to answer your question: yes. mostly. I don't know what you mean about assuming you know something about p(y>0). H[X|Y] >= the sum over all y> 0 of H[X|Y=y] $\endgroup$ Feb 17, 2020 at 13:52
  • $\begingroup$ Well $H[ X \mid Y, I] = \sum_{y} p(y,I=1) H[ X \mid Y = y, I=1 ] + \sum_{y} p(y,I=0) H[ X \mid Y = y, I=0 ]$. Isn't this the same as what I got above? What am I missing? $\endgroup$
    – csstudent
    Feb 18, 2020 at 5:30

1 Answer 1

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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).

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