= {\mathsf{P}}_{\mathcal{S}}\{\Theta_x({\mathcal{S}}) \subseteq A \mid \Theta_x({\mathcal{S}}) \neq \varnothing\},$$ where $A \subseteq \Theta$ is the assertion/hypothesis about $\theta$ of interest. @leafliu2012 give an alternative to conditioning on the event “$\Theta_x({\mathcal{S}}) \neq \varnothing$,” based on stretching, that tends to be more efficient.

The belief function is just one part of the inferential output. Since the belief function is sub-additive, i.e., ${\mathsf{bel}}_x(A;{\mathcal{S}}) + {\mathsf{bel}}_x(A^c;{\mathcal{S}}) \leq 1$, one actually needs both ${\mathsf{bel}}_x(A;{\mathcal{S}})$ and ${\mathsf{bel}}_x(A^c;{\mathcal{S}})$ to summarize the information in $x$ concerning the truthfulness of assertion $A$. In some cases, it is more convenient to report the *plausibility function* $$\label{eq:plausibility}
{\mathsf{pl}}_x(A;{\mathcal{S}}) = 1-{\mathsf{bel}}_x(A^c; {\mathcal{S}}).$$ Then the pair $({\mathsf{bel}}_x,{\mathsf{pl}}_x)(A;{\mathcal{S}})$ characterize the IM output. The IM and fiducial approaches both start with a representation of the sampling model using auxiliary variables, but, beyond that, there are some important differences. First, by taking the predictive random set ${\mathcal{S}}= \{U\}$, with $U \sim {\mathsf{P}}_U$, a random singleton, the corresponding IM is exactly the fiducial distribution for $\theta$. However, this singleton random set is not nested, hence, not admissible, so the desirable validity properties in Section \[SS:validity\] are not guaranteed. Second, a subtle point, the interpretation of probability changes as one proceeds along the fiducial argument. One starts with a 