and basic properties of IMs.

The discussion in @imbasics focuses on the case where $\theta$ and $U$ are of the same dimension. But there are many problems, e.g., iid data from scalar parameter models, where the dimension of the auxiliary variable is greater than that of the parameter. In such cases, efficiency can be gained by first reducing the dimension of the auxiliary variable to be predicted, though it is not obvious how this should be done in general. Here we focus our attention on an auxiliary variable dimension reduction step based on conditioning. The key observation is that, typically, certain functions of the auxiliary variables are fully observed. By conditioning on those observed characteristics of the auxiliary variable, we can effectively reduce the dimension of the unobserved characteristics to be predicted. A motivating example, demonstrating the efficiency gain from dimension reduction, along with the detailed developments are presented in Section \[S:conditional\]. The proposed dimension-reduction approach, based on conditioning, can be viewed as a general tool for combining information about $\theta$ across samples—a counterpart to Bayes’ theorem and sufficiency. With the lower-dimensional auxiliary variable, we proceed to construct what is called a *conditional IM*. We prove a validity theorem that 