is relatively easy. However, rarely does the model directly admit a scalar auxiliary variable representation. To see this, suppose $X_1,\ldots,X_n$ are independent ${{\sf N}}(\theta,1)$ with unknown mean $\theta$. In vector notation, an association is $X = \theta 1_n + U$, where $1_n$ is an $n$-vector of unity, and $U \sim {\sf N}_n(0,I)$. At first look, it seems that one must predict an $n$-dimensional auxiliary variable $U$. But efficient prediction of $U$ would be challenging, even for moderate $n$, so reducing its dimension—ideally to one—would be a desirable first step. After reducing the dimension to one, choosing efficient predictive random sets is as easy as in the scalar auxiliary variable case considered in @imbasics.

The basic point is that one pays a price, in terms of efficiency, for predicting higher-dimensional auxiliary variables. To see this better, we shall take a closer look at the normal mean problem above, with $n=2$. That is, we have a baseline association $$X_1 = \theta + U_1 \quad \text{and} \quad X_2 = \theta + U_2,$$ where $U_1,U_2$ are independent ${{\sf N}}(0,1)$. To make things simple, consider the following change of variables: $Y_1 = X_1+X_2$ and $Y_2 = X_1-X_2$. In the new variables, we have $$Y_1 = 