I'm stuck with the following "physical problem": I have a set $E=\{1, \ldots, n\}$, a function $f : E \to \mathbb{R}$ and a secret value $x_0\in E$. My goal is to find $x_0$.
Furthermore, I know that for $x\neq x_0$ (resp. $x= x_0$), $f(x)$ is supposed to be drawn from a fixed continuous distribution $D$ (resp. $D'$).
Hence my first thought was to sort all $x$ according to:
$$\Pr(x=x_0\mid\forall x'\in E, f(x'))$$
At this point I have already a problem, because I don't know how to formalize the intuitive meaning of $\Pr(\cdot\mid\forall x\in E,f(x))$ (I can't find any way to write the RHS as an event).
And then applying a kind of Bayes formula for continuous variables:
$$\Pr(x=x_0\mid\forall x'\in E, f(x'))=\frac{\text{Pr}(\forall x'\in E,f(x')\mid x=x_0) \Pr(x=x_0)}{\sum_{x''}\Pr(\forall x'\in E, f(x')\mid x''=x_0) \Pr(x''=x_0)}$$
Thus, as I think my model implies to choose $x_0$ uniformly at random in $E$, only $\Pr(\forall x'\in E,f(x')\mid x=x_0)$ depends on $x$. And my intuition tells me that if my model was correctly formalized, it could be expressed with the CDF $F_D$ and $F_D'$. How can I do this?
I'm not familiar with statistics, so I'm expecting an explanation from probability theory.
To have somewhat nicer notation, let $Y = (Y_1,\ldots,Y_n)$ be the random variables which you write as $f(1),\ldots,f(n)$, and let $X$ be the random variable which you write as $x_0$. Then, let $y = (y_1,\ldots,y_n)$ be your observations. We have for each $x \in E$ that \begin{align*} P(X=x \mid Y=y) &= \frac{f_Y(y \mid X = x)P(X =x)}{\sum_{x' \in E}f_Y(y \mid X = x')P(X=x')} \\\\ &= \frac{f_{D'}(y_x)\prod_{x' \in E,x'\neq x}f_D(y_{x'})}{\sum_{x' \in E} f_{D'}(y_{x'})\prod_{x'' \in E,x''\neq x'}f_D(y_{x''}),}\\\\ &= \frac{f_{D'}(y_x)}{f_D(y_x)\sum_{x' \in E}\frac{f_{D'}(y_{x'})}{f_D(y_{x'})}}\\\\ &= \frac{f_{D'}(y_x)}{f_D(y_x)}\Bigg/{\sum_{x' \in E}\frac{f_{D'}(y_{x'})}{f_D(y_{x'})}} \end{align*} In other words, the probability that $X = x$ is given by the ratio $f_{D'}(y_x)/f_D(y_x)$, "how many times more likely $D'$ is to give $y_x$ than $D$", with all these ratios normalized so that they add up to 1.