What About the Posterior Distributions When the Model is Non-dominated?

Abstract

Starting from the first inception of philosophical research that had subsequently led to subjective probability and Bayesian statistics, and to date the most recent developments, the probabilistic nature and the related statistical implications of Bayes theorem have been thoroughly discussed. However, the substantial contents of such a formula is very deep and new contributions are still continuing after 250 years. The simplest form of Bayes theorem is met when dominated statistical models are dealt with. This is, in a sense, comfortable, specially as far as parametric models are considered. Actually, most statistical techniques in the frame of parametric inference refer to dominated statistical models. Different problems in the applications, however, can lead to considering non-dominated models. In these cases, some complications and intriguing conclusions can arise. Concerning non-dominated statistical models, we devote this note to discussing some mathematical features that may sometimes escape the attention of statisticians. We deal with questions and results that, at a first glance, may appear of almost-exclusive measure-theoretic interest. However, they have a real statistical meaning of their own and the present note aims to stimulate some reflections about this field.