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

## TL;DR

This paper explores the relationship between profile likelihoods and marginal likelihoods, demonstrating their equivalence in normal models using the Jeffreys prior, and discusses the uniqueness of this case.

## Contribution

It shows that profile likelihoods can be true likelihoods in normal models through their equivalence to marginal likelihoods with Jeffreys prior.

## Key findings

- Profile likelihoods can be identical to marginal likelihoods in normal models.
- The Jeffreys prior achieves this equivalence.
- Such an equivalence is likely unique to normal models.

## Abstract

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.