A Bayesian analysis of the parameter vector, W (Wj = number of elements in category j), of a multivariate hypergeometric distribution is considered. It is shown that if, a priori, W is compound multinomial (CMtn), then, a posteriori, W is a translated CMtn. Many properties of the CMtn distribution are derived. These include joint moments of all orders; a characterization in terms of independent compound Poisson variables, conditional distribution of one subvector given another, and joint distributions of disjoint and overlapping sums of the components. It is shown that in applications related to analysis of variance and contingency tables, the parameters of interest are functions of W. A method is proposed for approximating the posterior distributions of such parameters, and a numerical example involving a performance index defined on a contingency table is presented.