I want to find the conjugate prior distribution for the Fisher's noncentral hypergeometric distribution? Basically I want to integrate the parameters out of the distributions to arrive at the Bayesian Likelihood of observing some data with the noncentral hypergeometric distribution.
Perhaps a conjugate prior does not exist for the noncentral hypergeometric distribution.
If someone wants to confirm this, it is worth noting that the conjugate to a univariate hypergeometric distribution is the beta-binomial; the conjugate to a multivariate hypergeometric distribution is the Dirichlet-Multinomial (from Fink, 1997). For more details on the derivation, see Dyer and Pierce, 1993.
Dyer, Danny and Pierce, Rebecca L. "On the Choice of the Prior Distribution in Hypergeometric Sampling". Communications in Statistics-Theory and Methods, 1993, 22(8), 2125-214
Fink, D. 1997. A Compendium of Conjugate Priors.
From the statement of the question, it seems as though you don't require conjugacy per se, rather you would like an analytical solution to your integration. From the form of the distribution, it would appear at first glance that the analytics of most solutions would be rather messy and difficult to interpret. The "analytic" solution would likely involve non-analytic functions (such as gamma function, beta function, confluent hypergeometric function), which may require numerical evaluation anyway.
May be quicker to use MCMC, rejection sampling or some other numerical technique to do the integration. But this means you need to choose a prior for you parameters (ideally, one which describes what you know about them).
One choice which comes to my mind, is the product of the beta binomial prior for the "central hypergeometric" part, and a beta distribution of the second kind for the odds ratio parameter. (beta distribution of the second kind is the distribution of the odds ratio of a beta distributed random variable, similar to a F distribution). But this is somewhat an arbitrary choice, only based on the conjugacy for the central hypergeometric distribution, and a "heavy-tailed" distribution (possibly robust? definitely more robust than gamma or inverse gamma) for the odds ratio parameter.
Also, which parameters are you integrating out? and which parameters are you taking the likelihood of?