we will give a general prescription for increasing efficiency by reducing the dimension. The key is that, in general, some functions of the original auxiliary variable are fully observed, like $V_2=U_1-U_2$ in this simple example. Then the strategy is to condition on what is fully observed to sharpen prediction of what is not observed. Since this “conditioning to sharpen inference” strategy is commonly used in statistics, similar considerations are natural in the IM framework.

Dimension reduction via conditioning {#SS:cim}
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Here we propose a conditioning strategy, whereby a simultaneous information aggregation and dimension reduction is achieved, that results in an overall gain in efficiency. The intuition is that some functions of the unobserved ${u^\star}$ are actually observed, so these characteristics do not need to be predicted. Focusing only on the unobserved characteristics of ${u^\star}$ leads directly to a dimension reduction. However, knowledge about the observed characteristics helps to better predict those unobserved characteristics, so information is accumulated and prediction is sharpened. The general strategy is as follows:

-   Identify an observed characteristic, $\eta(U)$, of the auxiliary variable $U$ whose distribution is free (or at least mostly free) of $\theta$, and

-   define a conditional association that relates an unobserved characteristic, 