We describe a hierarchical Bayesian approach for inference about a parameter $θ$ lower-bounded by $α$ with uncertain $α$, derive some basic identities for posterior analysis about $(θ,α)$, and provide illustrations for normal and Poisson models. For the normal case with unknown mean $θ$ and known variance $σ^2$, we obtain Bayes estimators of $θ$ 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.