Objective Priors for Discrete Parameter Spaces

The development of objective prior distributions for discrete parameter spaces is considered. Formal approaches to such development – such as the reference prior approach – often result in a constant prior for a discrete parameter, which is questionable for problems that exhibit certain types of structure. To take advantage of structure, this article proposes embedding the original problem in a continuous problem that preserves the structure, and then using standard reference prior theory to determine the appropriate objective prior. Four different possibilities for this embedding are explored, and applied to a population-size model, the hypergeometric distribution, the multivariate hypergeometric distribution, the binomial-beta distribution, the binomial distribution, and determination of prior model probabilities. The recommended objective priors for the first, third and fourth problems are new. AMS 2000 subject classification: Primary 62F15; secondary 62A01, 62B10, 62E20. Some key words: Parameter-based asymptotics, Binomial, Discrete parameter space, Hypergeometric, Jeffreys-rule priors, Prior model probabilities, Reference priors.