= Saddlepoint conditional likelihood adjustment
{title2=$B(\psi)=\tfrac12\log d_{\lambda\lambda}(\psi,\widehat\lambda_\psi)$}
For a regular two-parameter <exponential family>, condition on the <sample mean> of the nuisance canonical statistic. <Exponential tilting> to its observed value and the <saddlepoint density approximation> give a nuisance-statistic density proportional to $d_{\lambda\lambda}^{-1/2}$ times its large-deviation exponential. Dividing the joint <likelihood> by this density gives the <profile log-likelihood> plus the displayed adjustment, up to a data-only constant. The sign is positive. This construction assumes an interior saddle with positive curvature and a sufficiently regular nonlattice density; conditioning on a lattice statistic requires the corresponding mass approximation.
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