# reference request – Divergence between random variables after transformation

Let $$X$$ and $$Y$$ be random variables with laws $$mu_X$$, $$mu_Y$$ and $$d$$ be some $$f$$-divergence (e.g. KL, total variation, Hellinger). Writing $$d(X,Y)$$ for the divergence between $$mu_X$$ and $$mu_Y$$, are there known (upper or lower) bounds on $$d(g(X),g(Y))$$ in terms of $$g$$ and $$d(X,Y)$$?

Of course, the most natural candidate for these bounds would be an expression involving $$d(X,Y)$$: For example, by taking $$g$$ to be a constant, we know that there is no lower bound of the form $$d(X,Y)le Ccdot d(g(X),g(Y))$$, but maybe there is an upper bound.

(There is nothing really special about divergences here, and $$d$$ could be a metric such as Wasserstein. Divergences have a nice representation via densities, in which case the usual Jacobian transformation seems like an appealing tool, but I have not gotten far with this yet.)

Clarification: In the most general form, the bounds would of course also depend on $$f$$. I would be satisfied for interesting bounds that hold for one of the “usual” $$f$$ such as those outlined above.