The Plausibility and Likelihood Functions

Abstract

The notion of likelihood is an important concept in modern statistics. In particular, the likelihood ratio has been used by several authors [19, 37] to measure the strength of the evidence represented by observations in statistical problems. This idea works fine when the goal is to evaluate the strength of the available evidence for a simple hypothesis versus another simple hypothesis. However, the applicability of this idea is limited to simple hypotheses because the likelihood function is primarily defined on points, i.e. simple hypotheses, of the parameter space. In this chapter, we extend the notion of likelihood from simple to composite hypotheses. It turns out that this extended notion of likelihood is the plausibility function of the hint derived from a generalized functional model. This allows us to define a general weight of evidence that is applicable to both simple and composite hypotheses. Classical distribution models introduced in the previous chapter do not convey enough information to generate a natural and general concept of weight of evidence. In other words, they are too coarse a representation of the mechanical process underlying the generation of the data observed. Generalized functional models and the theory of hints are the appropriate tools for defining such a general concept. Furthermore, a weight of evidence can be given a concrete significance by finding a well understood and simple situation leading to the same weight of evidence. The relevant concepts and ideas are explained by means of a familiar urn problem in addition to the general analysis of a real-world medical problem.