On the Foundations of Statistical Inference

TLDR

Conditional experimental frames of reference are central to statistical inference, a concept highlighted by Fisher, Cox, Tukey, and others, and supported by Fisher, Barnard, and Bayesian scholars. The authors formalize conditionality as a principle, derive the likelihood principle from it, and discuss its mathematical consequences and connections to Bayesian methods. Adopting the likelihood principle leads to intrinsic significance and confidence levels and intrinsic standard errors, thereby revising standard inference methods.

Abstract

Abstract The concept of conditional experimental frames of reference has a significance for the general theory of statistical inference which has been emphasized by R. A. Fisher, D. R. Cox, J. W. Tukey, and others. This concept is formulated as a principle of conditionality, from which some general consequences are deduced mathematically. These include the likelihood principle, which has not hitherto been very widely accepted, in contrast with the conditionality concept which many statisticians are inclined to accept for purposes of “informative inference.” The likelihood principle states that the “evidential meaning” of experimental results is characterized fully by the likelihood function, without other reference to the structure of an experiment, in contrast with standard methods in which significance and confidence levels are based on the complete experimental model. The principal writers supporting the likelihood principle have been Fisher and G. A. Barnard, in addition to Bayesian writers for whom it represents the “directly empirical” part of their standpoint. The likelihood principle suggests certain systematic reinterpretations and revisions of standard methods, including “intrinsic significance and confidence levels” and “intrinsic standard errors,” which are developed and illustrated. The close relations between non-Bayesian likelihood methods and Bayesian methods are discussed.