Justifying Bayesianism by Dynamic Decision Principles

As yet, no general agreement has been reached on whether the Bayesian or the frequentist (Neyman-Pearson, NP) approach to statistics is to be preferred. Whereas Bayesians adhere to coherence conditions of de Finetti, Savage, and others, frequentists do not consider these conditions normative and deliberately and knowingly violate them. Hence further arguments, bringing more clarity on the disagreements, are warranted. Providing such arguments, by refining the coherence conditions, is the purpose of this paper. It invokes recent arguments from the economic literature demonstrating that some seemingly self-evident principles for dynamic decision making have a surprising implication for static decisions: They imply Bayesianism. These principles are forgone-event independence (independence of past counterfactual events, often called consequentialism in decision theory and known as the conditionality principle in statistics), dynamic consistency (what is optimal at some given time point is independent of the time point at which that is decided), and two other conditions. Thus, a more sensitive diagnostic tool is obtained for identifying the disagreements between Bayesians and frequentists. If a frequentist does not mind violating Bayesian coherence, a Bayesian can now ask a follow-up question: Which of the dynamic principles will the frequentist give up? The debate may lead either to Bayesianism or to better implementations of non-Bayesian models in dynamic decision situations and to better non-Bayesian methods for updating information. The diagnostic tool sheds new light on NP hypothesis testing. NP theory requires that statistical procedures are laid down before data are observed. It adheres to dynamic consistency but violates forgone-event independence. Forgone-event independence, however, is so natural that NP practitioners adhere to it and observe the data before deciding on a statistical procedure. They are thus led into violations of dynamic consistency.