# Maximization of information over set of non-injective functions

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 non-injective deterministic functions.

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 non-injective functions. We have $$F_1 \supset F_2$$.
• Yes, you are right. But one question: Let $X$ also be independent from $Z$, do we then have equality? Aug 16, 2022 at 7:56