Let $X\to Y\to Z$ be three random variables.
The data processing inequality states $I(X;Y)\geq I(X;Z)$.
Further assume $Y=f(X)$ where $f:\mathcal{X}\to\mathcal{Y}$ is an arbitrary function.
What more can we say about how $I(Y;Z)=I(f(X);Z)$ relates to $I(X;Y)=I(X;f(X))$ and $I(X;Z)$?
I.e. can one somehow add the mutual informations along the path or obtain an inequality relating the three pairwise mutual informations? Somehow the choice of $f$ establishes an upper bound on what information can potentially be shared between $X$ and $Z$, but how does it affect the mutual information $I(Y;Z)=I(f(X);Z)$?