Let $X, Y, Z$ be Polish spaces; $M$ a collection of full-support Borel measures on $X$; $\nu$ a Borel measure on $Y$; $f:X\times Y \to Z$ continuous with the property that $f(\cdot,y)$ is injective for every $y\in Y$.
Suppose $(x,y)\in X\times Y$ is drawn according to $\mu\otimes\nu$ for some $\mu \in M,$ and $z=f(x,y).$ Under what conditions on $(M,\nu,f)$ will the distribution of $x$ conditional on $z$ be (a.s.-$z$) some $\mu' \in M$?
- A concrete example: take $X=Y=Z=\mathbb R,$ $M$ the class of Gaussian measures, $\nu$ Gaussian, and $f$ linear.
- A trivial case: $M$ is the set of all full-support Borel measures, and $f(X,y)$ is the same for every $y\in Y$.
- Another trivial case: $f(x,\cdot) = g \enspace\forall x\in X,$ for some $g:Y\to Z$.
I'd be very interested in a general answer to this, but I'm ultimately interested in some nice examples. I'd love non-Gaussian examples for which each of $X,Y,Z$ is either $\mathbb R$ or $[0,\infty),$ $M$ is a collection nicely expressed by 1 or 2 parameters, and all measures in sight admit continuous densities.
Any thoughts are very appreciated!