Prior-Free Probabilistic Inference
Ryan Martin, Chuanhai Liu · 2015
It is difficult to discuss various attempts to solve the problem of statistical inference, in part because they all have slightly different goals and there is no agreed-upon criteria that can allow one to conclude that one approach is “correct” or better than all the others. The main goal of this chapter is to introduce, in Section 2.2, what, in our opinion, are the essentials for prior-free probabilistic inference. A first conclusion is that the classical frequentist methods, which are focused on the design of procedures with good sampling properties, are not “inference” in our sense. Then, in Section 2.3, we delve into a critical survey of the existing methods which are both prior-free, in the sense that no real prior information is needed to carry out the necessary calculations, and probabilistic, in the sense that the inferential output is a sort of probability distribution. This includes Bayesian methods based on default non-informative priors, Fisher’s fiducial inference, the Dempster-Shafer theory of belief functions, and a number of other approaches. Having identified the shortcomings of these existing approaches, we can then address two important questions. First, in Section 2.4, we discuss the role of probability in statistical inference. Our conclusion is that the familiar probability measures are not the appropriate tool for summarizing evidence in data for or against an assertion of interest, and we propose the use of the distribution of a random set, i.e., a belief function, as a more appropriate tool. Second, in Section 2.5, we give a high-level explanation of how our new way of thinking can lead to improvements to the existing approaches.