 Modified profile likelihood

For parameter of interest $\psi$, constrained nuisance fit ${\hat{\lambda }}_{\psi }$, and suitable ancillary sample coordinates $(\hat{\psi },\hat{\lambda },a)$, this adjustment includes both nuisance observed information and a sample-space Jacobian determinant. The derivative holds $\hat{\psi },a$ fixed. Implicit differentiation of the constrained nuisance score function gives the equivalent expression ${\ell }_p+\frac{1}{2}\log |j_{\lambda \lambda }|-\log |{\ell }_{\lambda ;\hat{\lambda }}|$. The semicolon distinguishes data-coordinate differentiation from parameter differentiation. Omitting the Jacobian requires an additional justified simplification; the information adjustment alone is not universal.