admissible, given $h$, with distribution ${\mathsf{P}}_{{\mathcal{S}}|h}$ as in . If $\Theta_{T(x)}({\mathcal{S}}) \neq \varnothing$ with ${\mathsf{P}}_{{\mathcal{S}}|h}$-probability 1 for all $x$ such that $H(x)=h$, then the conditional IM is conditionally valid, i.e., for any $A \subseteq \Theta$, $$\label{eq:cond.valid.bel}
\sup_{\theta \not\in A} {\mathsf{P}}_{X|\theta}\{{\mathsf{cbel}}_{T(X)|h}(A;{\mathcal{S}}) \geq 1-\alpha \mid H(X) = h\} \leq \alpha, \quad \forall \; \alpha \in (0,1).$$

Now is a good time to recall Remark \[re:separable\]. More general decompositions of the baseline association are allowed in the discussion in Section \[SS:cim\], but only for the “separable” version is it possible to prove a conditional validity theorem. The point is that a condition like $c(X,U)=0$ does not identify a fixed subset of the sample space on which probability calculations can be restricted—the subspace would depend on $U$.

Since the calibration property in Theorem \[thm:cond.valid\] holds for all assertions $A$, we may translate to a statement in terms of the corresponding plausibility function: $$\label{eq:cond.valid.pl}
\sup_{\theta \in A} {\mathsf{P}}_{X|\theta}\{{\mathsf{cpl}}_{T(X)|h}(A;{\mathcal{S}}) \leq \alpha \mid H(X)=h\} \leq \alpha, \quad \forall\;\alpha \in (0,1).$$ So, in addition to providing an objective scale for interpreting the conditional belief and plausibility function values, provides desirable properties of conditional IM-based frequentist procedures. For example, if $h=H(x)$ is observed, the conditional $100(1-\alpha)\%$ plausibility region for 