particular, if we write this association as $$\label{eq:assoc}
X = a(\theta,U), \quad U \sim {\mathsf{P}}_U.$$ Throughout, subscripts on ${\mathsf{P}}$ indicate which quantity is random.

Fiducial inference employs the sampling distribution ${\mathsf{P}}_U$ *after* $X=x$ is observed. The IM approach is different in that it treats the unobserved value of $U$, which is tied to the observed data $X=x$ and the *true value* of $\theta$, as the fundamental quantity. Then the goal is to predict this unobserved value with a random set before conditioning on $X=x$ and inverting . Let $({\mathbb{U}},\mathscr{U},{\mathsf{P}}_U)$ be a probability space, where $\mathscr{U}$ is rich enough to contain all closed subsets of ${\mathbb{U}}$. Take a collection ${\mathbb{S}}$ of closed (hence ${\mathsf{P}}_U$-measurable) subsets of ${\mathbb{U}}$, assumed to contain $\varnothing$ and ${\mathbb{U}}$. This collection will serve as the support of the predictive random set. We shall also assume that the collection ${\mathbb{S}}$ is nested, i.e., either $S \subseteq S'$ or $S' \subseteq S$ for all $S,S' \in {\mathbb{S}}$. Now define the predictive random set ${\mathcal{S}}\sim {\mathsf{P}}_{\mathcal{S}}$, supported on ${\mathbb{S}}$, with distribution ${\mathsf{P}}_{\mathcal{S}}$ satisfying $${\mathsf{P}}_{\mathcal{S}}\{{\mathcal{S}}\subseteq K\} = \sup_{S \in {\mathbb{S}}: S \subseteq K} {\mathsf{P}}_U(S), \quad K \subseteq {\mathbb{U}}.$$ Predictive random sets with these properties are called *admissible*. @imbasics give a sort 