Frameworks for prior-free posterior probabilistic inference

TL;DR

This paper critically examines prior-free probabilistic inference methods, highlighting the limitations of fiducial inference and proposing the inferential model (IM) framework as a valid, efficient alternative demonstrated through benchmark examples.

Contribution

It clarifies the distinction between prior-free and non-prior-free methods and introduces the IM framework as a genuinely prior-free approach for valid and efficient inference.

Findings

Fiducial inference is not truly prior-free.

The IM framework is valid and efficient for probabilistic inference.

Demonstrated IM methods on benchmark problems like the bivariate normal and Behrens--Fisher.

Abstract

The development of statistical methods for valid and efficient probabilistic inference without prior distributions has a long history. Fisher's fiducial inference is perhaps the most famous of these attempts. We argue that, despite its seemingly prior-free formulation, fiducial and its various extensions are not prior-free and, therefore, do not meet the requirements for prior-free probabilistic inference. In contrast, the inferential model (IM) framework is genuinely prior-free and is shown to be a promising new method for generating both valid and efficient probabilistic inference. With a brief introduction to the two fundamental principles, namely, the validity and efficiency principles, the three-step construction of the basic IM framework is discussed in the context of the validity principle. Efficient IM methods, based on conditioning and marginalization are illustrated with two benchmark examples, namely, the bivariate normal with unknown correlation coefficient and the Behrens--Fisher problem.