I have a problem that hasn't yet been addressed, although follows similar lines of reasoning here and here. My problem is as follows:

I have $N$ urns, each with a different number of black and white marbles. Denote the $i^{th}$ urn, $u_i$ having $n_i$ black marbles, and $N_i - n_i$ white marbles.

What I want is to know the minimum number of samples I should take (without replacement) from each urn such that I have a "reasonable estimate" of how many white marbles I have in total. Let's replace "reasonable estimate" with 95% credible intervals. And as I sample more and more my credible intervals decrease. Assume urns are independent of each other.

My first attempt was to model $n_i$ as a Hypergeometric, so that I can use a Beta-Binomial conjugate prior, which yields a Beta-Binomial posterior, per this answer. And I was thinking that I could just take samples from the posterior to develop a credible interval for each urn. And perhaps multiply them together in some way (since they are independent).

But my problem is that I don't know what the actual distribution of white to black marbles are in each urn. I want to come up with an estimate for this based on a small sample.

Can anyone provide some insight as to how I can approach this problem?

  • $\begingroup$ For some intuition, please note that you don't necessarily need a good credible interval for each urn. Urns with small values of $N_i$ and urns with very concentrated priors on $n_i$ will contribute relatively little uncertainty. This suggests that you begin by writing some expression for the uncertainty (however you want to measure it) for any particular set of sample sizes and outcomes from the urns; you can then seek to minimize its expectation (relative to your prior) subject to a constant sum of non-negative sample sizes. $\endgroup$ – whuber Nov 2 '18 at 19:32
  • $\begingroup$ @whuber Unfortunately I don't follow, but for simplicity let's ignore the Bayesian case and assume I just wanted to model the likelihood function for a single urn. I'd use a Hypergeometric right? But as far as I know, the parameters of a HG are N (total number of marbles), K (total number of black marbles) and n (sample size). I'm suggesting that I don't know what K is. I want to know how big n should be such that my 95% confidence interval for K is within some width. Does that make sense? $\endgroup$ – ilanman Nov 2 '18 at 20:22
  • $\begingroup$ Thinking about this more, perhaps all I need is to write down the MLE for a Hypergeometric with unknown K and perform whatever algorithm for finding K. And for multiple urns I can just add the Ks since they are independent. $\endgroup$ – ilanman Nov 3 '18 at 12:54

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