# Upper Bound on Mutual Information of a Function of Random Variables

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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