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

Inference for a constrained parameter in presence of an uncertain constraint