From Likelihood to Plausibility

Several authors have explained that the like­ lihood ratio measures the strength of the evi­ dence represented by observations in statisti­ cal problems. This idea works fine when the goal is to evaluate the strength of the avail­ able evidence for a simple hypothesis versus another simple hypothesis. However, the ap­ plicability of this idea is limited to simple hypotheses because the likelihood function is primarily defined on points simple hy­ potheses of the parameter space. In this paper we define a general weight of evidence that is applicable to both simple and compos­ ite hypotheses. It is based on the Dempster­ Shafer concept of plausibility and is shown to be a generalization of the likelihood ratio. Functional models are of a fundamental im­ portance for the general weight of evidence proposed in this paper. The relevant con­ cepts and ideas are explained by means of a familiar urn problem and the general analysis of a real-world medical problem is presented.