= Modified profile likelihood
{title2=$\ell_m=\ell_p-\tfrac12\log|j_{\lambda\lambda}|-\log|\partial\widehat\lambda_\psi/\partial\widehat\lambda|$}
= Barndorff-Nielsen modified profile likelihood
{c}
{synonym}
For parameter of interest $\psi$, constrained nuisance fit $\widehat\lambda_\psi$, and suitable ancillary sample coordinates $(\widehat\psi,\widehat\lambda,a)$, this adjustment includes both nuisance <observed information> and a sample-space <Jacobian determinant>. The derivative holds $\widehat\psi,a$ fixed. Implicit differentiation of the constrained nuisance <score function> gives the equivalent expression $\ell_p+\tfrac12\log|j_{\lambda\lambda}|-\log|\ell_{\lambda;\widehat\lambda}|$. The semicolon distinguishes data-coordinate differentiation from parameter differentiation. Omitting the Jacobian requires an additional justified simplification; the information adjustment alone is not universal.
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