José Miguel Bernardo

In multi-parameter models, reference priors typically depend on the parameter or quantity of interest, and it is well known that this is necessary to produce objective posterior distributions with optimal properties. There are, however, many situations where one is simultaneously interested in all the parameters of the model or, more realistically, in functions of them that include aspects such as prediction, and it would then be useful to have a single objective prior that could safely be used to produce reasonable posterior inferences for all the quantities of interest. In this paper, we consider three methods for selecting a single objective prior and study, in a variety of problems including the multinomial problem, whether or not the resulting prior is a reasonable overall prior.

Project Euclid - mathematics and statistics online

This article considers the development of objective prior distributions for discrete parameter spaces. 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, and the binomial distribution. The recommended objective priors for the first, third, and fourth problems are new.

The complete final product of Bayesian inference is the posterior distribution of the quantity of interest. Important inference summaries include point estimation, region estimation and precise hypotheses testing. Those summaries may appropriately be described as the solution to specific decision problems which depend on the particular loss function chosen. The use of a continuous loss function leads to an integrated set of solutions where the same prior distribution may be used throughout. Objective Bayesian methods are those which use a prior distribution which only depends on the assumed model and the quantity of interest. As a consequence, objective Bayesian methods produce results which only depend on the assumed model and the data obtained. The combined use of intrinsic discrepancy, an invariant information‐based loss function, and appropriately defined reference priors, provides an integrated objective Bayesian solution to both estimation and hypothesis testing problems. The ideas are illustrated with a large collection of non‐trivial examples.

Project Euclid - mathematics and statistics online