# What is the exact definition of profile likelihood?

Does anyone here know the exact definition of Profile Likelihood? Or does it have one?

I would suggest

Next, I am going to summarise the definition of the Profile or maximised likelihood.

Let $\theta$ be a vector parameter that can be decomposed as $\theta = (\delta,\xi)$, where $\delta$ is a vector parameter of interest and $\xi$ is a nuisance vector parameter. This is, you are interested only on some entries of the parameter $\theta$. Then, the likelihood function can be written as

where $f$ is the sampling model. An example of this is the case where $f$ is a normal density, $y$ consist of $n$ independent observations, $\theta=(\mu,\sigma)$ and say that you are interested on $\sigma$ solely, then $\mu$ is a nuisance parameter.

The profile likelihood of the parameter of interest is defined as

Sometimes you are also interested on a normalised version of the profile likelihood which is obtained by dividing this expression by the likelihood evaluated at the maximum likelihood estimator.

You can find an example with the normal distribution here.

I hope this helps.