  We describe a hierarchical Bayesian approach for inference about a parameter
$\theta$ lower-bounded by $\alpha$ with uncertain $\alpha$, derive some basic
identities for posterior analysis about $(\theta,\alpha)$, and provide
illustrations for normal and Poisson models. For the normal case with unknown
mean $\theta$ and known variance $\sigma^2$, we obtain Bayes estimators of
$\theta$ that take values on $\mathbb{R}$, but that are equally adapted to a
lower-bound constraint in being minimax under squared error loss for the
constrained problem.
