Consensus priors for multinomial and binomial ratios

Abstract

Reference analysis is an objective Bayesian approach to finding non-informative prior distributions. For various models involving nuisance parameters the reference prior has been shown to be superior to the multiparameter Jeffreys prior. In this article, the performance of the reference prior is evaluated for models that are defined at the extremes of parameter ranges, in which case reference and Jeffreys priors are shown to be potentially informative. Two quantities of interest are analyzed: the ratio of two multinomial parameters and the ratio of two independent binomial parameters, where the latter is just one example of inference based on the common 2 × 2 contingency table. For these two specific examples, we show that reference and/or Jeffreys priors lead to overinformative inference in extreme data situations that would seem common in, for example, many medical contexts when interest is in rare events. It is recommended that in the context of noninformative priors and binary data with extreme observed data, practitioners adopt priors for which prior predictive distributions are uniform, as per Bayes’s original argument.