On Resolving the Controversy in Statistical Inference

Abstract

Summary 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 R 1, 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 non-coherence 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.