dimension the same as that of $\theta$. In addition to providing a sort of summary of the data, like in the classical context, this auxiliary variable dimension reduction has a unique advantage in the IM context: efficient predictive random sets for the lower-dimensional $V_T$ are easier to construct. Furthermore, the conditioning aspect sharpens our predictive ability, improving efficiency even more. We witnessed, empirically, these gains in efficiency in the simple example in Section \[SS:normal.example\]. Some further remarks on this conditional association, and its connections to Fisher’s sufficiency and Bayes theorem, are collected in Section \[SS:remarks\].

Once a decomposition is available, construction of the corresponding IM follows exactly as in Section \[S:intro\]. To simplify the presentation later on, here we restate the three-step construction of a *conditional IM*.

Associate $T(x)$ and $\theta$ with the new auxiliary variable $v_T = \tau(u)$ to get the collection of sets $\Theta_{T(x)}(v_T) = \{\theta: T(x) = b(v_T,\theta)\}$, $v_T \in {\mathbb{V}}_T$, based on .

Fix $h=H(x)$. Predict the unobserved value ${v^\star}_T$ of $V_T$ with a *conditionally admissible* predictive random set ${\mathcal{S}}\sim {\mathsf{P}}_{{\mathcal{S}}|h}$ (see Section \[SS:condcred\]).

Combine results of the A- and P-steps to get $$\label{eq:condfocal}
\Theta_{T(x)}({\mathcal{S}}) = \bigcup_{v_T \in {\mathcal{S}}} \Theta_{T(x)}(v_T) \subseteq \Theta.$$ Then the corresponding conditional 