Here's the problem: A probabilistic function $f$ takes as its argument an input $x \in X$ with vector output $\mathbf {y} = f(x)$, where other inputs from $X$ might also produce $\mathbf y$, but likely with different probabilities. Now, using only one's knowledge of the function and the output, how can one determine the likelihood that $x$ was indeed the input? That is, what steps should one take to create a formula that can calculate this likelihood?
I only have a vague notion in my head about (somehow) generating a probability distribution for random vector $\boldsymbol {Y} = f(\mathbf {X}), \forall x \in X$, then doing something with those results. If, for example, a different input $w \in X$ yielded a probability distribution that was $0$ at some indexes where $\mathbf y$ was not $0$, then $w$ could be ruled out. But of course, that's just an easy case.
A solution or references to fitting concepts or areas of mathematics would be great. Tag suggestions might also be helpful.
