2
$\begingroup$

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.

$\endgroup$

1 Answer 1

0
$\begingroup$

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 $F_1 \supset F_2$.

$\endgroup$
1
  • $\begingroup$ Yes, you are right. But one question: Let $X$ also be independent from $Z$, do we then have equality? $\endgroup$
    – Cesare
    Aug 16, 2022 at 7:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.