Normed likelihood as saddlepoint approximation

Barndorff-Nielsen's formula (normed likelihood with constant-information metric) has been proffered as an approximate conditional distribution for the maximum-likelihood estimate, based on likelihood functions. Asymptotic justifications are available and the formula coincides with the saddlepoint approximation in full exponential models. It is shown that the formula has wider application than is presently indicated, that in local analysis it corresponds to Laplace's method of integration, and that it corresponds more generally to a saddlepoint approximation.