Profile log-likelihood (${\ell }_p(\psi )=\ell (\psi ,{\hat{\lambda }}_{\psi })$)

The logarithm of the profile likelihood is the full log-likelihood evaluated at the nuisance fit constrained to the specified parameter of interest. It preserves likelihood ratios and maximizers, but is not generally a normalized sampling log-density. Conditioning and integrating out nuisance parameters are different operations.