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The Problem

Given probability distributions $P(\theta)$ and $P(X)$, and given an inverse function $Y=f^{-1}(X,\theta)$ that returns a unique $Y$. How can one estimate the unkown distribution $P(Y)$ in the following hierarchical model?

$\theta = f(X,Y)$

$Y \sim P(\cdot)$

$X \sim P(\cdot)$

The Questions

  • What is this problem called, is it a defined problem within statistics with a name?
  • What methods exist (or could be reasonably proposed) to solve for $P(Y)$?

Current Thoughts

At first it seems like a solution is to simply sample independently from $P(\theta)$ and $P(X)$ to estimate $P(Y)$ using the function $Y=f^{-1}(X,\theta)$, but that would be incorrect: the hierarchical model above implies a dependence between $X$ and $\theta$ such that one would need to instead sample from the joint distribution $P(\theta,X)$ (which is not given).

It seems like this problem may be related to importance sampling or approximate Bayes computation.

Improve this question
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  • $\begingroup$ This is a generalized deconvolution problem. (if $f(X,Y)=X+Y$ then it is exactly deconvolution) $\endgroup$
    – J. Delaney
    Commented May 18, 2023 at 16:49
  • $\begingroup$ I do not think the concept of "estimation" applies in that case since this is not a statistical problem. Mathematically, assuming $\theta$ and $X$ are independent, the change of variable formula provides the distribution of $Y$ since $f$ is invertible wrt its first entry. Simulation-wise, generating $(\theta,X)$ produces exact generations of $Y$. If the joint distribution of $(\theta,X)$ is unknown, there is no answer to the question. $\endgroup$
    – Xi'an
    Commented May 24, 2023 at 8:05

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