An epistemic interpretation of the posterior likelihood ratio distribution

Abstract

Often the expression of a likelihood ratio involves model parameters θ. This fact prompted many researchers to argue that a likelihood ratio should be accompanied by a confidence interval, as one would do when estimating θ itself. We first argue against this, based on our view of the likelihood ratio as a function of our knowledge of the model parameters, rather than being a function of the parameters themselves. There is, however, another interval that can be constructed, and which has been introduced in the literature. This is the interval obtained upon sampling from the so-called ‘posterior likelihood ratio distribution’, after removing, say, the most extreme 5% of a sample from this distribution. Although this construction appears in the literature, its interpretation remained unclear, as explicitly acknowledged in the literature. In this article we provide an interpretation: the posterior likelihood ratio distribution tells us which likelihood ratios we can expect if we were to obtain more information. As such, it can play a role in decision making procedures, for instance about the question whether or not it is worthwhile to try to obtain more data. The posterior likelihood ratio distribution has no relevance for the evidential value of the current data with our current knowledge. We illustrate all this with a number of examples.