I've got this problem where basically:
$$ Z = f(X) \sim Normal(0, 1) $$
$f$ is pretty non-linear and I don't have its inverse function. In practice (using Stan's MCMC samplers or Tensorflow's optimisers or by using literally any method), is it possible to draw samples from X?
My limited progress:
I don't really know how I'd specify such a model in Stan, or if it's even possible ($f$ is non-linear, almost certainly non-monotonic so I can't add a Jacobian adjustment and treat X as a parameter).
I dunno if treating X as a variable and constructing an objective that applies $f$ to X and optimises in the direction that makes the sample "more normal" is the way to go here. I don't think that such an approach would scale well, with increasing dimensionality of X and Z.
I'm looking for any other ideas on how to sample from X. Is it possible to use the KL divergence here somehow?