belief and plausibility functions are given by $$\label{eq:condbelief}
\begin{split}
{\mathsf{cbel}}_{T(x)|h}(A;{\mathcal{S}}) & = {\mathsf{P}}_{{\mathcal{S}}|h}\{\Theta_{T(x)}({\mathcal{S}}) \subseteq A \mid \Theta_{T(x)}({\mathcal{S}}) \neq \varnothing\} \\
{\mathsf{cpl}}_{T(x)|h}(A;{\mathcal{S}}) & = 1-{\mathsf{cbel}}_{T(x)|h}(A^c; {\mathcal{S}}). 
\end{split}$$ These functions can be used for inference on $\theta$ just like those in Section \[S:ims\]. The notation ${\mathsf{cbel}}$ and ${\mathsf{cpl}}$ is meant to indicate that these are belief and plausibility functions, depending on $(T,H)(x)$, based on predicting the lower-dimensional auxiliary variable $V_T = \tau(U)$ in the conditional association .

When a decomposition is available, the conditional association and the corresponding conditional IM analysis is intuitively reasonable. One could ask, however, if there is any loss in using IMs built from instead of . The following proposition establishes that there is no loss.

\[prop:conditioning\] Suppose the baseline association admits a decomposition of the form . Let ${\mathcal{S}}$ be a predictive random set for $U$ in the baseline association with the property that ${\mathsf{P}}_{\mathcal{S}}\{\Theta_x({\mathcal{S}}) \neq \varnothing\} > 0$ for all $x$. Then there exists a predictive random set ${\mathcal{S}}_{H(x)}$ for $\tau(U)$, depending on ${\mathcal{S}}$ and $H(x)$, in the conditional association such that ${\mathsf{bel}}_x(A;{\mathcal{S}}) = {\mathsf{cbel}}_{T(x)}(A; {\mathcal{S}}_{H(x)})$ for all $x$ and all assertions $A \subseteq \Theta$.

See Appendix \[S:proofs\] for the proof. This says that the conditional 