SELECTING AND SHARPENING INFERENCES IN SIMULTANEOUS INFERENCE WITH A BAYESIAN APPROACH

Abstract

A frequentist simultaneous confidence interval procedure requires the predetermination of the comparisons and their corresponding forms of confidence intervals before viewing the data in order that the error probability is controlled at a preassigned level. This often renders it less sensitive to detecting actual true differences and may result in it including many noninformative inferences. On the other hand, by taking a Bayesian approach, we can select the comparisons of interest and construct corresponding joint credible intervals after having viewed the data. This enables us to focus on those significant differences of interest and consequently to be able to make sharper inferences. The joint posterior probability of the credible intervals play a similar role as the joint coverage probability of the simultaneous confidence intervals, that is, to guarantee, with at least that probability, all the inferences made using the intervals are correct at the same time. In this article, we consider some standard problems in simultaneous inference and discuss how a Bayesian approach may be implemented. The methodologies are illustrated with examples.

Keywords:

• Simultaneous inference
• Multiple comparisons
• Simultaneous confidence intervals
• Bayesian inference
• Posterior distribution
• Joint credible intervals
• Gibbs sampler

ACKNOWLEDGMENTS

We thank the referee and the Associate Editor for critical and constructive comments.