  Many functions have approximately-known upper and/or lower bounds,
potentially aiding the modeling of such functions. In this paper, we introduce
Gaussian process models for functions where such bounds are (approximately)
known. More specifically, we propose the first use of such bounds to improve
Gaussian process (GP) posterior sampling and Bayesian optimization (BO). That
is, we transform a GP model satisfying the given bounds, and then sample and
weight functions from its posterior. To further exploit these bounds in BO
settings, we present bounded entropy search (BES) to select the point gaining
the most information about the underlying function, estimated by the GP
samples, while satisfying the output constraints. We characterize the sample
variance bounds and show that the decision made by BES is explainable. Our
proposed approach is conceptually straightforward and can be used as a plug in
extension to existing methods for GP posterior sampling and Bayesian
optimization.
