I am struggling to understand the following. Let $X_1$, $W_1$, and $Z_1$ be mutually independent discrete random variables with finite alphabets, $S_1 := X_1 + W_1$, and $f(\cdot)$ be some deterministic scalar function.

Is it true that $I[f(S_1)+Z_1;X_1] \leq I[S_1+Z_1;X_1]$, where $I[\cdot]$ denotes mutual information?

I have tried a number of things but I have not been able to show it. The function $f(S_1)$ is a clustering-type of function, i.e., it groups elements of the alphabet of $S_1$ and assigns the average of them as an element of the alphabet of $f(S_1)$.

Any hint would be very appreciated it. Cheers, Carlos.


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