Are profile likelihoods likelihoods? No, but sometimes they can be

We offer our two cents to the ongoing discussion on whether profile likelihoods are “true” likelihood functions, by showing that the profile likelihood function can in fact be identical to a marginal likelihood in the special case of normal models. Thus, profile likelihoods can be “true” likelihoods insofar as marginal likelihoods are “true” likelihoods. The prior distribution that achieves this equivalence turns out to be the Jeffreys prior. We suspect, however, that normal models are the only class of models for which such an equivalence between maximization and marginalization is exact.