On Resolving the Controversy in Statistical Inference

Abstract

The controversy concerning the fundamental principles of statistics still remains unresolved. It is suggested that one key to resolving the conflict lies in recognizing that inferential probability derived from observational data is inherently noncoherent, in the sense that their inferential implications cannot be represented by a single probability distribution on the parameter space (except in the Objective Bayesian case). More precisely, for a parameter space RI, the class of all functions of the parameter comprise equivalence classes of invertibly related functions, and to each such class a logically distinct inferential probability distribution pertains. (There is an additional cross-coherence requirement for simultaneous inference.) The noncoherence of these distributions flows from the nonequivalence of the relevant components of the data for each.Noncoherence is mathematically inherent in confidence and fiducial theory, and provides a basis for reconciling the Fisherian and Neyman-Pearsonian viewpoints. A unified theory of confidence-based inferential probability is presented, and the fundamental incompatibility of this with Subjective Bayesian theory is discussed.

Fiducial inference in the pivotal family of distributions

“…This example is also studied by Wilkinson [19] . Note that the fiducial distribution also has a mass with size …”

“…One can see that in refs. [9,[18][19][20][30][31]. Therefore Definition 5 is applied to the situation in which the fiducial inferences have a frequentist meaning, such as the regular parametric function given in Definition 6.…”

“…Many discussions on them, including some controversies, can be found in refs. [11][12][13][14][15][16][17][18][19][20][21][22][23]. The views of Efron [24] about the fiducial inference should be noted.…”

Le choix bayésien

“…Comme nous le détaillons dans ce chapitre, la formule de Bayes est apparue pour la première fois en 1761, dans le cadre de l'exemple binomial de la Section 1.2, exposé par le révérend Thomas Keynes (1921), le début du XXème siècle fut surtout marqué par, tout d'abord, Kolmogorov, qui proposa dans les années 1920 une axiomatisation de la théorie des probabilités semblant contredire le paradigme bayésien et la notion de probabilité subjective, ensuite par Fisher qui s'éloigna de l'approche bayésienne (Fisher, 1912) en définissant la fonction de vraisemblance (Fisher, 1922), puis en développant la Statistique fiduciaire (Fisher, 1930), et qui ne révisa jamais son opinion négative sur la Statistique bayésienne. Cette opposition paraît quelque peu paradoxale, car la Statistique fiduciaire tentait, en un certain sens, de surmonter la difficulté de choisir une loi a priori en la construisantà partir de la fonction de vraisemblance (Seidenfeld, 1992), dans le même esprit que les approches non informatives de Jeffreys (1939) et Bernardo (1979 Stein, 1959, Wilkinson, 1977, et les références dans Zabell, 1992. Le livre de Jeffreys (1939) est le premier traité moderne de Statistique bayésienne : il couvre, en plus de la notion d'a priori non informatif, celles de loi prédictive, de facteur de Bayes et d'a priori impropre.…”

Book Reviews

“…263-264, but does not solve it. Perhaps it is a pity that Lad does not refer to the attempt of Wilkinson (1977) to develop a non-coherent but workable theory of inference. In the long term, population ecology may perhaps provide some assistance (e.g.…”