Conditional likelihood ($L_c(\theta )=f_{\theta }(y\mid S=s)$)

A conditional likelihood evaluates the conditional distribution of the observed data given a specified statistic. If that distribution does not involve a nuisance parameter, it permits inference about the remaining parameter without estimating the nuisance parameter. The conditioning statistic need not be ancillary; possible information loss must be assessed within the model. Conditional maximum likelihood maximizes this function.

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.