Let $\mathcal{A}$ and $\mathcal{B}$ be two sets such as $|\mathcal{A}| = n$ and $|\mathcal{B}| = m$.
Let $f$ be a function such that \begin{align*} f_x \colon \mathcal{A} &\to \mathcal{B}\\ a &\mapsto f_x(a). \end{align*}
For each $x \in \mathcal{A}$ we a compute the probability distribution of couples $(a,f_x(a))$ noted $D_x$.
Suppose that for a fixed $x$ that is unknow, we run an experiment that returns a set of samples $\mathcal{S} = \{(a,f_x(a))|a \in \mathcal{A}\}$
Given $\mathcal{S}$ and the probability distributions $D_x$, how to compute the probability $P(X=x)$ ?
For a single couple (i.e. $|\mathcal{S}| = 1$), the probability $P(X=x|\mathcal{S})$ can be easily computed from all the distributions $D_x$ but how to deal with larger sets?