are profile likelihoods likelihoods?!
A recent arXived paper by Oliver J. Maclaren is asking this very question. And argue for a positive answer. One of the invoked sources is Murray Aitkin’s integrated likelihood book, which I criticised here and elsewhere. With the idea of the paper being that
“….there is an appropriate notion of integration over variables that takes likelihood functions to likelihood functions via maximization.”
Hmm…. The switch there is to replace addition with maximisation, probability with possibility, and… profile likelihood as marginal possibility under this new concept. I just do not see how adapting these concepts for the interpretation of the profile likelihood makes the latter more meaningful, since it still overwhelmingly does not result from a distribution density at an observed realisation of a random variable. This reminds me a paper I refereed quite a long while ago where the authors were using Schwarz’ theory of distributions to expand the notion of unbiasedness. With unclear consequences.
Xi'an's Og 
August 16, 2018 at 3:11 am
[…] turns out that you can (arguably) think of likelihood in terms of possibility, rather than probability, theory, and […]
March 29, 2018 at 5:41 am
The issue of course depends on what you want from a likelihood. Perhaps it could be made clearer that the idea is to explain how ‘pure likelihood’ theory can be viewed as a self-consistent instance of possibilitic reasoning. That is, maxitive measure theory.
I’d argue that many people use profile likelihoods like this in practice but often have no formal theory to ‘justify’ what they already know to be reasonable.
Fisher said many years ago that likelihood is not an additive function nor a set function. Basu argued why not both? This approach allows extension to set functions but drops additivity.
Why should uncertainty be additive? Especially if all models are wrong.
March 28, 2018 at 8:26 pm
XLM provides a very balanced discussion of many of the nuances – maybe agreeing with Berger et al or maybe stopping somewhat short of that https://arxiv.org/pdf/1010.0810.pdf
Keith O’Rourke
March 27, 2018 at 12:44 pm
Thanks Christian for commenting this issue which was much discussed in some areas of inference eg variance components in mixed models.
To me there is a reference paper in this field by berger, liseo and wolpert 1999 statistical science , 14, 1-28 making clear the differences between all these subproducts of likelihood.
The profile is for sure not a likelihood as you said as not generated from a data distribution .
For instance in linear mixed models the score of the profile likelihood as a function of variance components has no longer a zero expectation.