I am working on a problem using a multivariate hypergeometric likelihood. The multivariate hypergeometric distribution does not belong to the exponential family of distributions, so (to my knowledge) we cannot guarantee that a conjugate prior exists.
However, Wikipedia claims on its page about conjugate priors that the univariate hypergeometric distribution is conjugate with a beta binomial. This post shows that the posterior in the univariate case is for M-x, the "number of target individuals in the population shifted by the number observed in the sample".
Does the multivariate hypergeometic distribution have a conjugate prior? My hunch is that it would be the multivariate form of the beta-binomial, the dirichlet-multinomial. However, I don't know how to show that analytically. If it does have a conjugate prior, what would the posterior's hyper-parameters be?