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Assuming there is a set $S$ that you are given subsets of, $s_1, s_2, ..., s_n$, estimate $|S|$ (and a confidence interval if possible) making as few assumptions as possible.

I'm not going to quibble over assuming the subsets are selected in an independent or uncorrelated way.

If you can assume every element in $S$ has an equal chance of being selected and $n=2$:

$|S| \simeq \frac{|s_1| |s_2|}{|s_1 \cap s_2|}$

(Think of $s_1$ and $s_2$ as the lower and left portion of a square.)

But what about $n>2$ or each element has a different (but non zero) chance of being selected?

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If elements may have different chances of being selected, hardly anything can be said (there might be many elements wirh probability 0). –  Hagen von Eitzen Sep 28 '12 at 18:29
1  
I don't know about estimation theory, but [set-theory] it is not. –  Asaf Karagila Sep 28 '12 at 18:29
    
@HagenvonEitzen: then it would not be possible to avoid assumptions about the distribution or probabilities. –  BCS Sep 28 '12 at 19:24

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