Let $X$, $Y$, $Z$ be discrete random variables, with $Y$ and $Z$ independent. Does the following equality hold **if $Z$ is independent also of $X$**?
$$
\max\_{f\_{Y,Z}} \big\{ \ I(X; f\_{Y,Z}(Y,Z)) \ \big\} = \max\_{f\_X, f\_Y} \big \{ \ I(X; f\_Y(Y), f\_Z(Z)) \ \big \}
$$
where the maximization is taken over all **non-injective**, deterministic functions.


P.S.: See [this](https://mathoverflow.net/questions/426330/maximization-of-information-over-set-of-non-injective-functions) for the inequality version of the question.



Maximization of information over set of non-injective functions (Equality)
No. Let $Y,Z$ be iid Bernoulli(1/2), and let $X=Y+Z$ mod 2, which induces pairwise independence. The only non-injective deterministic functions $f\_Y,f\_Z$ are constants, rendering the RHS zero. For the LHS, we can take $f\_{Y,Z}(Y,Z)$ equal to the the binary AND of $Y$ and $Z$, which is not injective, and not independent of $X$. Hence, the LHS is something positive, showing there can be strict inequality.


