Inherent Difficulties of Non-Bayesian Likelihood-based Inference, as Revealed by an Examination of a Recent Book by Aitkin

TL;DR

This paper critically examines Aitkin's likelihood-based inference approach, highlighting its incompatibility with Bayesian principles and questioning its practical relevance in statistical analysis.

Contribution

It provides a detailed analysis of the limitations of non-Bayesian likelihood inference as presented in Aitkin's recent book, emphasizing conceptual conflicts with Bayesian methods.

Findings

The approach conflicts with Bayesian probabilistic interpretation.

It is incompatible with standard Bayesian inference principles.

The method has limited applicability in practical statistical analysis.

Abstract

For many decades, statisticians have made attempts to prepare the Bayesian omelette without breaking the Bayesian eggs; that is, to obtain probabilistic likelihood-based inferences without relying on informative prior distributions. A recent example is Murray Aitkin's recent book, {\em Statistical Inference}, which presents an approach to statistical hypothesis testing based on comparisons of posterior distributions of likelihoods under competing models. Aitkin develops and illustrates his method using some simple examples of inference from iid data and two-way tests of independence. We analyze in this note some consequences of the inferential paradigm adopted therein, discussing why the approach is incompatible with a Bayesian perspective and why we do not find it relevant for applied work.