Invited Discussion of J.O. Berger: Four Types of Frequentism and Their Interplay with Bayesianism

One of the merits of this far reaching article is to show that not all “Frequentisms” are equal. Furthermore that there are frequentist approaches which are compelling scientifically, notably the “Empirical Frequentist” (EP), which can be paraphrased as “The proof of the pudding is in the eating”. Somewhat surprisingly to some (but anticipated in Wald’s admissibility Theorems in Decision Theory), is the conclusion that the easiest and best way to achieve the EP property is through Bayesian reasoning, perhaps more exactly, through Objective Bayesian reasoning. (I am avoiding the expression Empirical Bayesian reasoning which would be appropriate if it wasn’t associated with a very particular group of methods. It is argued below that a better name would be “Bayes Empirical”) I concentrate on Hypothesis Testing since that is the most challenging area of deeper disagreement among schools. From this substantive classification of Frequentisms, emerges the opportunity for a convergence, which is even more satisfying than a compromise, between schools. This may only be fully achieved if the prior probabilities are known, which is not usually the case. However, particularly in Hypothesis Testing, prior probabilities can and should be estimated and its uncertainty acknowledged in a Bayesian way. This may be termed perhaps, Bayes Empirical: The systematic empirical study of Prior Possibilities based on relevant data, acknowledging its uncertainty.