not can be obtained by slight extension of the argument in the previous remark. This important application of conditional IMs deserves further investigation. See, also, [@xie.singh.2012].

Validity of conditional IMs {#SS:condcred}
---------------------------

Here we extend the validity results in @imbasics to the conditional IM context. The main obstacle is that the distribution function ${\mathsf{P}}_{\mathcal{S}}$, determined by the conditional distribution ${\mathsf{P}}_{V_T|H(x)}$ in , depends on data through the value $H(x)$. This is handled in Theorem \[thm:cond.valid\] below by conditioning on the observed value of $H(X)$.

Fix $h \in H({\mathbb{X}})$, and let ${\mathbb{S}}_h$ be a collection of closed ${\mathsf{P}}_{V_T|h}$-measurable subsets of ${\mathbb{V}}_T$ that contains both $\varnothing$ and ${\mathbb{V}}_T$. Like before, we also assume that ${\mathbb{S}}_h$ is nested in the sense that either $S \subseteq S'$ or $S' \subseteq S$ for all $S,S' \in {\mathbb{S}}_h$. Then ${\mathcal{S}}$ is a *conditionally admissible* predictive random set, given $h$, if its distribution ${\mathsf{P}}_{{\mathcal{S}}|h}$ satisfies $$\label{eq:cond.natural.measure}
{\mathsf{P}}_{{\mathcal{S}}|h}\{{\mathcal{S}}\subseteq K\} = \sup_{S \in {\mathbb{S}}_h: S \subseteq K} {\mathsf{P}}_{V_T|h}\{S\}, \quad K \subseteq {\mathbb{V}}_T.$$ In this case, the distribution of ${\mathcal{S}}$ depends on the particular $h$. We now have the following extension of the validity theorem to the case of conditional IMs.

\[thm:cond.valid\] For any $h$, suppose that ${\mathcal{S}}$ is conditionally 