  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.
