Let $X,Y$ be continuous random variables with $X$ defined over $\mathcal{A}$, and let $f: \mathcal{A} \to \mathcal{A}$, $g: \mathcal{A} \to \mathcal{A}$ be any functions. Is it true that


$$
I(Y;f \circ g(X) ) \leq I(Y; f(X))
$$


where $I(\cdot\,; \cdot)$ denotes mutual information. Note that
data processing inequality gives,
$$
I(Y; f\circ g(X)) \leq I(Y; g(X)) \leq I(Y; X)\\
I(Y; f(X)) \leq I(Y;X)
$$



Data processing inequality
There is no reason for that. For instance, one can choose the functions $f$ and $g$ in such a way that $g$ is a bijection on the range of $X$, whereas $f$ is constant on the range of $X$ and is a bijection on the range of $g(X)$. Then $I(Y,f(X))=0$, whereas $I(Y,f\circ g(X))=I(Y,X)$.


