On the definition of likelihood function

TL;DR

This paper proposes a general, measure-theoretic definition of likelihood functions using Radon-Nikodým derivatives, emphasizing its importance in complex models and Bayesian analysis, and clarifying the role of dominating measures.

Contribution

It introduces a broad, rigorous framework for likelihood functions applicable to infinite-dimensional models and clarifies the significance of choosing appropriate dominating measures.

Findings

Likelihood functions are proportional under different dominating measures.

Continuous versions of densities are crucial for the Likelihood Principle.

Using the predictive measure as a dominating measure is beneficial in Bayesian analysis.

Abstract

We discuss a general definition of likelihood function in terms of Radon-Nikod\'{y}m derivatives. The definition is validated by the Likelihood Principle once we establish a result regarding the proportionality of likelihood functions under different dominating measures. This general framework is particularly useful when there exists no or more than one obvious choice for a dominating measure as in some infinite-dimensional models. We discuss the importance of considering continuous versions of densities and how these are related to the Likelihood Principle and the basic concept of likelihood. We also discuss the use of the predictive measure as a dominating measure in the Bayesian approach. Finally, some examples illustrate the general definition of likelihood function and the importance of choosing particular dominating measures in some cases.