$U_1$ is conditionally independent given $U_0$. Here ${\mathsf{P}}_{U_0}$ is like the prior, and the distribution induced by $u_1 \mapsto a(\theta,u_1)$ given $U_0=\theta$ determines the likelihood. It is clear that the function $a(U_0,U_1)$ is fully observed, so the conditional IM strategy would employ the conditional distribution of $U_0$ given the observed value $x$ of $a(U_0,U_1)$. It is not hard to see that the belief function based on the “naive” predictive random set ${\mathcal{S}}=\{U_0\}$ is exactly the Bayesian posterior distribution function. So in any problem with a known prior distribution, the Bayes solution can be obtained as a special case of the conditional IM. No non-naive predictive random set is needed here because the naive IM itself is valid; this is consistent with the simple corresponding fact for posterior probabilities under a Bayes model with known prior.

\[re:more.bayes\] As a follow-up to Remark \[re:bayes\], since a full prior is not required to construct a conditional IM, it is possible to develop an inferential framework based on conditional IMs and “partial prior information.” For example, valid prior information may be available for some but not all components of $\theta$. Incorporating the prior information where it is available while remaining prior-free where it is 