On the mathematical foundations of likelihood theory

Abstract

We discuss a general definition of likelihood function in terms of Radon-Nikodým 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 also discuss some versions of densities which are specially important when obtaining the likelihood function. In particular, we argue in favor of continuous versions of densities and highlight how these are related to the basic concept of likelihood. Finally, we present a method, based on the concept of differentiation of measures, to obtain a valid likelihood function, i.e., which is in accordance with the Likelihood Principle. Some examples are presented to illustrate the general definition of likelihood function and the importance of choosing particular dominating measures in some cases.