Identifiability of structural characteristics: How relevant is it for the Bayesian approach?

Abstract

The role of identification in the Bayesian approach is still debatable. Since Lindley [Bayesian Statistics. A Review (1971) Philadelphia], most Bayesian statisticians pretend that unidentifiabiity causes no real difficulty in their approach. Recently, Wechsler, Izbicki and Esteves [Amer. Statist. 67 (2013) 90–93] provide a simple example illustrating this perspective. By critically reading Wechsler, Izbicki and Esteves [Amer. Statist. 67 (2013) 90–93], we intend to show that the Bayesian approach is far from being free of the identification problems, provided that the interest is focused on the interpretation of the parameters. It is written using a rather ancient style, the so-called Platonic dialogues. In modern times, there are beautiful examples of that, particularly in Foundations of Mathematics, where debatable subjects are discussed: let us refer Heyting [Intuitionism. An Introduction (1971) North-Holland Publishing Company], where the debate between a formalist and an intuitionist is presented as a dialogue; or Lakatos [Proofs and Refutations. The Logic of Mathematical Discovery (1976) Cambridge University Press], where the relationship between proofs and conjectures is magnificently illustrated. We hope that this style will help to understand why identifiability really matters in the Bayesian approach.