Let $X,Y$ be continuous random variables and $f,g$ 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, $$ I(Y; f\circ g(X)) \leq I(Y; g(X)) \leq I(Y; X)\\ I(Y; f(X)) \leq I(Y;X) $$

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

Let $X,Y$ be continuous random variables and $f,g$ be any functions. Is it true that

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

Let $X,Y$ be continuous random variables and $f,g$ 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, $$ I(Y; f\circ g(X)) \leq I(Y; g(X)) \leq I(Y; X)\\ I(Y; f(X)) \leq I(Y;X) $$