I encountered a sampling problem and managed to approximate programmatically. However, I'm interested in determining the analytical solution.
There are two sets $A$ and $B$. $|A|$ is known, $|B|$ is unknown.
Each element in $A$ mapped into not empty subset of $B$.
There is a sample $\mathbb{S}$. Where $A_s$ and $B_s$ are known, the mapping between $a_s where \space a_s \in A_s$ and $B_s' \space where \space B_s' \subset B_s$ is also known.
We should estimate the expected value of $|B|$.
Are there enough details in this problem for it to be solved analytically, or will some assumptions be necessary?
An example of a real-life application of this problem, just for illustration. In some social network we have 1000 accounts that held by $x, x\le1000$ individuals. We detected that 800 accounts are held by 400 individuals. The question is - what is the expected value of x.
We could naively figure out that if 400 individuals hold 800 accounts, each holds (on average) 2 accounts, so there are 500 individuals for 1000 accounts. There is definitely a mistake here - we assume that additional 200 accounts are held by new individuals, which is incorrect - additional accounts may belong to individuals already known to us.