Let $X$, $Y$, $Z$ be discrete random variables, with $Y$ and $Z$ independent. Does the following equality hold? $$ \max_{f_{Y,Z}} \big\{ \ I(X; f_{Y,Z}(Y,Z)) \ \big\} \le \max_{f_X, f_Y} \big \{ \ I(X; f_Y(Y), f_Z(Z)) \ \big \} $$ where the maximization is taken over all noninjective deterministic functions.
1 Answer
It seems the inequality should be held in the opposite way due to the fact that the function family on the left that you are taking maximum over is a superset of the function family on the right. More specifically, let $F_1:=\{(y,z) \to f_1(y,z)\}$ and $F_2:= \{(y,z) \to (f_y(y),f_z(z))\}$ where $f_1,f_y,f_z$ are arbitrary noninjective functions. We have $F_1 \supset F_2$.

$\begingroup$ Yes, you are right. But one question: Let $X$ also be independent from $Z$, do we then have equality? $\endgroup$– CesareAug 16, 2022 at 7:56