of complete-class theorem for admissible predictive random sets. In scalar $\theta$ problems, ${\mathsf{P}}_U $ is often ${{\sf Unif}}(0,1)$, so an important example of an admissible predictive random set is $$\label{eq:default.prs}
{\mathcal{S}}= \{u: |u-\tfrac12| \leq |U-\tfrac12|\}, \quad U \sim {{\sf Unif}}(0,1).$$ @imbasics [Corollary 1] show that this ${\mathcal{S}}$—called the “default” predictive random set—has a variety of good properties, and these good properties often carry over to the corresponding IM. It should be mentioned that admissible predictive random sets are not unique. In this paper we focus primarily on simplicity, but the ideas on optimal predictive random sets in @imbasics can also be applied here.

The following three steps—association, predict, and combine—define an IM.

Associate $X$, $\theta$, and $U \sim {\mathsf{P}}_U$, consistent with the sampling distribution $X \sim {\mathsf{P}}_{X|\theta}$, such that, for all $(x,u)$, there is a unique subset $\Theta_x(u) = \{\theta: x=a(\theta,u)\} \subseteq \Theta$, possibly empty, containing all possible candidate values of $\theta$ given $(x,u)$.

Predict the unobserved value $u^\star$ of $U$ associated with the observed data by an admissible predictive random set ${\mathcal{S}}$.

Combine ${\mathcal{S}}$ and the association $\Theta_x(u)$ specified in the A-step to obtain $$\label{eq:new.focal}
\Theta_x({\mathcal{S}}) = \bigcup_{u \in {\mathcal{S}}} \Theta_x(u).$$ Then compute the *belief function* $$\label{eq:belief}
{\mathsf{bel}}_x(A; {\mathcal{S}}) 