association is as good a starting point as the baseline association in the sense that any belief function that obtains from the latter can be matched by one that obtains from the former. Therefore, the best conditional IM can be no worse than the best baseline IM. However, as in Section \[SS:normal.example\], by working with predictive random sets for the lower-dimensional auxiliary variable in the conditional association, efficiency can be improved. The point of this paper, in fact, is that the best conditional IM is more efficient than the best baseline IM.

A shortcoming of Proposition \[prop:conditioning\] is that the predictive random set ${\mathcal{S}}_{H(x)}$ constructed in the proof may not be valid for prediction of $\tau(U)$, which prevents us from making a direct efficiency comparison of conditional versus baseline IMs. However, in the special case where $\tau(U)$ and $\eta(U)$ are independent, validity of that predictive random set obtains.

\[co:indep.cond\] In the setup of Proposition \[prop:conditioning\], suppose that $\tau(U)$ and $\eta(U)$ are independent. Then the predictive random set ${\mathcal{S}}'=\tau({\mathcal{S}})$ for $\tau(U)$ is valid and ${\mathsf{bel}}_x(A;{\mathcal{S}}) = {\mathsf{cbel}}_{T(x)}(A; {\mathcal{S}}')$ for all $x$ and all assertions $A \subseteq \Theta$.

Since the predictive random set for $\tau(U)$ is valid, a requirement for IMs, in 