The conditional score function is found to be generally fully informative concerning a parameter of interest when the conditioning statistic $S$ is sufficient for the nuisance parameter and has an exponential family distribution. Information is here measured by assuming the nuisance parameter to have been generated by an unknown mixing distribution and then computing the minimal Fisher information. The solution depends upon a study of the geometry of centered likelihood ratios within the space of zero-unbiased functions of $S$. The two-by-two table model is considered in detail.
"Efficiency of the Conditional Score in a Mixture Setting." Ann. Statist. 11 (2) 486 - 497, June, 1983. https://doi.org/10.1214/aos/1176346155