Ancillary-conditioned score density ($f_U(u\mid a;\theta )\approx {\sum }_{v:u(v,a)=u}\frac{p^{\ast }(v\mid a;\theta )}{|{\partial }_{v}u(v,a)|}$)

Fix an ancillary statistic $a$ and a parameter value at which the score function is evaluated. Express that score as a function $u(v,a)$ of the varying maximum-likelihood estimator coordinate $v$. The ordinary density change-of-variables formula applied to the P-star approximation gives the displayed density; several monotone branches require a sum. Its derivative is a sample-coordinate derivative, not generally the parameter observed information.