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Frameworks for prior-free posterior probabilistic inference

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.