conditions are more elusive. To keep the presentation simple, we shall mostly ignore these technicalities.

Validity of IMs {#SS:validity}
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Given ${\mathcal{S}}$, the corresponding IM is *valid for* $A$ if the belief function satisfies $$\label{eq:bel.valid}
\sup_{\theta \not\in A} {\mathsf{P}}_\theta\bigl\{ {\mathsf{bel}}_X(A;{\mathcal{S}}) \geq 1-\alpha \bigr\} \leq \alpha, \quad \forall\;\alpha \in (0,1).$$ The IM is simply called *valid* if it is valid for all $A$. In other words, the IM is valid for $A$ if ${\mathsf{bel}}_X(A;{\mathcal{S}})$ is stochastically no larger than ${{\sf Unif}}(0,1)$ when $X \sim {\mathsf{P}}_{X|\theta}$ with $\theta \not\in A$. That is, if $A$ is false, then the amount of support in data $X$ for $A$ will be large only for a relatively small proportion of $X$ values. @imbasics [@imbasics.c] show that this validity property holds whenever the predictive random set ${\mathcal{S}}$ is admissible. If the IM is valid for all $A$, then can be equivalently stated in terms of the plausibility function: $$\label{eq:pl.valid}
\sup_{\theta \in A} {\mathsf{P}}_{X|\theta}\{{\mathsf{pl}}_X(A; {\mathcal{S}}) \leq \alpha\} \leq \alpha, \quad \forall \; \alpha \in (0,1).$$ This formulation is occasionally more convenient than .

There are two important consequences of the validity theorem. First, it helps determine an objective scale on which the belief probabilities can be interpreted. Unlike 