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I have a finite set $X$ of unknown size. From that I independently draw a multiset of samples $S\subset X$. For each sample $s\in S$ I can give the exact probability $p(s)$ with which it has been drawn from $X$ such that $\sum_{x\in X}p(x)=1$ and how often it has been drawn. I also have a lower bound $l\le p(s)$ for the probability of each sample.

Given this information, how can I estimate the size of $X$?

A mark and recapture approach is not going to work because $|S| \ll |X|$ by several orders of magnitude. I have drawn about a billion of samples so far without encountering any duplicates.

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    $\begingroup$ Statement not clear (to me anyhow). Perhaps look at mark-recapture $\endgroup$ – BruceET Oct 8 at 16:13
  • $\begingroup$ @BruceET The population $X$ is much larger than the sample size I can reasonably achieve. I have so far drawn a few billion samples without any duplicates, so I don't think mark-recapture is feasible. What part of the statement is unclear? $\endgroup$ – fuz Oct 8 at 16:20
  • $\begingroup$ "For each sample $S∈s$ I can give the probability $p(S)$ it has to be drawn from $X$ such that ..." Maybe give examples from your work. How do you get the probabilities? Exact or estimated? $\endgroup$ – BruceET Oct 8 at 16:28
  • $\begingroup$ @BruceET I can compute these probabilities exactly. I am working on the same problem this question is about (sampling vertices of a graph that have a given distance from a fixed vertex). Once I found a vertex with the right distance, I can exactly compute the probability with which it was found. $\endgroup$ – fuz Oct 8 at 16:33
  • $\begingroup$ @BruceET Also, I'm sorry for the wrong spelling and grammar. I have fixed the post. $\endgroup$ – fuz Oct 8 at 16:35
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DISCLAIMER: Simulation has shown that this is not a good estimator but it might be helpful for further ideas

I propose the following estimator: With sample space $S$ and sample $s_1, ..., s_n$, we could use $\widehat{\#S}= n \times\frac{1}{\sum_{i=1}^np(s_i)}$ as an estimator for $\#S$

The theoretical idea is the following:

We have a random Variable $X$ that has the probability values $p_s:=p(s)$ as the sample space and a distribution, such that $P(X=p_s)=p(s)$, i.e. the numeric value of the sample and its probability coincide.

We then estimate the mean of this Random Variable $X$, i.e. $E[X]=\sum_{s\in S}p(s)^2$ by taking the sample mean $\sum_{i=1}^np(s_i)$ In addition to that we use the fact that $nE[X]=1 \iff n = \frac{1}{E[X]}$

So we use an unbiased estimator for $E[X]$ and plug it into the formula $n=\frac{1}{E[X]}$ (which means that the estimator for $\#S$ is not necessarily unbiased anymore.

Maybe we could also estimate the distribution of $X$ in order to get a correction-factor of our estimate that takes into consideration how much $1/E[X]$ deviates from $E[1/X]=n$.

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  • $\begingroup$ I do draw individual elements. $S\subset X$ is the set of samples I draw of which the probability $p(s)$ for each $s\in S$ is known. Here, $p(s)$ is the probability of drawing $s$ from $X$. $\endgroup$ – fuz Oct 8 at 20:02
  • $\begingroup$ Ok then the last paragraph is unnecessary but the rest might still help you. I will simulate this procedure tomorrow to check how well it works and update my post. $\endgroup$ – Sebastian Oct 8 at 20:07
  • $\begingroup$ I actually thought about something similar. We do know about the distribution of probabilities because we sampled them, so I guess there should be a way to make that exact. $\endgroup$ – fuz Oct 8 at 20:32
  • $\begingroup$ Re your formula: although this statistic is an estimator, what properties does it have? Why should it be any good at all? I don't follow your remarks about "uniformly distributed," because if that assumption is true, then all you have to do is compute a single probability $p_i$ and recover the population size as $1/p_i.$ $\endgroup$ – whuber Oct 8 at 21:09
  • $\begingroup$ As in the first line: "This is not a known method (at least to me) but just an idea that makes sense to me and might be worth exploring." $\endgroup$ – Sebastian Oct 8 at 21:12

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