valid IMs, the output from default-prior Bayesian, fiducial, and Dempster–Shafer inference does not have a specified scale for interpretation. For example, is a Bayesian or fiducial posterior probability of 0.9 a large value? It is common to think on the usual frequency scale, i.e., betting on an event with 0.9 probability wins 90% of the time, but there is no justification for this without some notion of validity as in or . Second, validity justifies the use of IM output to construct frequentist decision procedures with control on error rates. For example, a $100(1-\alpha)$% plausibility region for $\theta$ is given by $$\label{eq:plaus.region}
\{\theta: {\mathsf{pl}}_x(\theta; {\mathcal{S}}) > \alpha\}.$$ It follows easily from that this plausibility region has nominal $1-\alpha$ coverage probability. But we should emphasize here that, although plausibility functions can be used to construct frequentist procedures, the interpretation is quite different. For example, the plausibility region is understood as the collection of $\theta$’s such that each is *individually* sufficiently plausible, given $X=x$. Confidence/credible regions, on the other hand, say nothing about the plausibility of any particular $\theta$ they contain.

Conditional IMs {#S:conditional}
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Motivation {#SS:normal.example}
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In the case of a scalar auxiliary variable, construction of efficient predictive random sets 