Is profile likelihood a true likelihood? An argument in favor

Profile likelihood is the key tool for dealing with nuisance parameters in likelihood theory. It is often asserted, however, that profile likelihood is not a 'true' likelihood. One implication is that likelihood theory lacks the generality of e.g. Bayesian inference, wherein marginalization is the universal tool for dealing with nuisance parameters. Here we argue that profile likelihood has as much claim to being a true likelihood as a marginal probability has to being a true probability distribution. The crucial point we argue is that a likelihood function is naturally interpreted as a maxitive possibility measure: given this, the associated theory of integration with respect to maxitive measures delivers profile likelihood as the direct analogue of marginal probability in additive measure theory. Thus, given a background likelihood function, we argue that profiling over the likelihood function is as natural (or as unnatural, as the case may be) as marginalizing over a background probability measure. The connections to Bayesian inference can also be further clarified with the introduction of a suitable logarithmic distance function, in which case the present theory can be naturally described as 'Tropical Bayes' in the sense of tropical algebra.