Strong Likelihood Principle: Strengthening a Principle or Misunderstanding the Likelihood Function

Abstract

The strong likelihood principle (SLP) is conventionally derived from the sufficiency principle and a conditionality principle in an argument due to Birnbaum, and much of the literature contests whether the derivation is sound. We take a different approach. We ask what the SLP says when its terms are read carefully, and argue that the principle as ordinarily stated reflects a confusion about the domain of the likelihood function. The likelihood is naturally defined as a function on a family of distributions $M$, not on a parameter space, and once it is so defined the SLP collapses into its weak counterpart, the weak likelihood principle. The diagnosis is illustrated by analogy with monetary value, developed concretely through a comparison of the binomial and negative binomial families that share a parameter, and connected to the geometric structure of $M$ through the Fisher information metric. The same standardization emerges from a statistical argument about comparing measurements across populations and from a geometric argument about manifold distance; this convergence supplies the positive content of the weak likelihood principle.