1
$\begingroup$

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?

$\endgroup$
3
  • 2
    $\begingroup$ What are the parameters of the multivariate hypergeometric on which you want to use a conjugate prior? Using the standard definition, with the $K_i$'s being the parameters there is no conjugate prior. $\endgroup$
    – Xi'an
    Nov 1, 2021 at 17:26
  • $\begingroup$ The $K_i$'s would be the parameters. Is there a simple explanation as to why? Or has one just not been found yet? $\endgroup$
    – Ryan Folks
    Nov 1, 2021 at 19:41
  • $\begingroup$ As you wrote, this distribution is not from an exponential family, hence cannot enjoy a conjugate prior. $\endgroup$
    – Xi'an
    Nov 1, 2021 at 19:42

1 Answer 1

2
$\begingroup$

According to this compendium of conjugate priors, the prior for the multivariate hypergeometric distribution's parameters is a Dirichlet-Multinomial distribution.

(The fact that the hypergeometric distribution does not belong to the exponential family does not preclude the possibility for conjugate prior. Simply, its existence is not guaranteed.)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.