Confidence, credibility and prediction

This paper examines, from a historical and model-based Bayesian perspective, two inferential issues: (1) the relation between confidence coverage and credibility of interval statements about model parameters; (2) the prediction of new values of a random variable. Confidence and credible intervals have different properties. This worries some statisticians, who want them to have the same properties, in frequentist repeated sampling coverage and Bayesian credibility content. Research continues on the conditions under which intervals have the same repeated sampling coverage and credibility content. We conclude that these two inferential approaches generally have incommensurable properties, which converge only asymptotically in sample size, and in a restricted class of samples. The repeated confusion, by new students of statistics, of the coverage of a confidence interval with the credibility of the observed interval, suggests that in general the credibility of the interval is the more important inferential aim. We show how the credibility of a confidence interval can be assessed generally. Bayesian prediction of new values is well established using the posterior predictive distribution. This has some curious features, known since Laplace but not well understood. The prediction of new Bernoullis highlights these features. We suggest that the credibility of predictive intervals needs to be reassessed.