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The beta-binomial distribution characterizes the number of successes in $n$ trials, but where the probability of success at each trial is unknown or random. However, suppose that you had finite populations and required that samples occurred without replacement (hypergeometric distribution).

Is there an analogue to the beta-binomial distribution which characterizes sampling without replacement from a population of $n$ items, where $b$ are labelled black and $w$ are labelled white ($n = b + w$), but where you do not precisely know what the values of $b$ and $w$ are?

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  • $\begingroup$ Are you sampling without replacement from a single population and you are looking for a distribution for the remaining unsampled items, or are you sampling without replacement from several identical populations and looking for a distribution for the whole population? Note that in the same way that the beta family provides conjugate distributions for a binomial likelihood, the beta-binomial family provides conjugate distributions for a hypergeometric likelihood $\endgroup$ – Henry Dec 6 '18 at 8:06
  • $\begingroup$ @Henry The former. sampling $k$ items from a population of black and white marbles, whose precise number I do not know, but where I may know the distribution. $\endgroup$ – ux74bn1 Dec 6 '18 at 16:00

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