# How can I estimate a population size from the probability of randomly drawn members?

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.

• Statement not clear (to me anyhow). Perhaps look at mark-recapture – BruceET Oct 8 at 16:13
• @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? – fuz Oct 8 at 16:20
• "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? – BruceET Oct 8 at 16:28
• @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. – fuz Oct 8 at 16:33
• @BruceET Also, I'm sorry for the wrong spelling and grammar. I have fixed the post. – fuz Oct 8 at 16:35

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 $$E[1/X]=n$$.

• 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 $X$. – fuz Oct 8 at 20:02
• 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. – Sebastian Oct 8 at 20:07
• 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. – fuz Oct 8 at 20:32
• 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 $1/p_i.$ – whuber Oct 8 at 21:09
• 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." – Sebastian Oct 8 at 21:12