Suppose we have a conditional density $p(X|Y)$, and that we know $X = f(Z)$ for some function $f$ and a random variable $Z$.
Without the conditionals, we know that $p(X) = p(f^{-1}(X))|\det J(X)|$ where $J$ is the Jacobian of the inverse transform $f^{-1}$ by applying the change of variables theorem.
I am stuck on (maybe overthinking it)
a) applying the change of variables to the conditional density $p(X|Y)$
b) the "reverse" case of applying it to $p(Y|X)$
Any references or explanations would be great