Description
Due to their flexibility, Gaussian processes (GPs) have been widely used in nonparametric function estimation. A prior information about the underlying function is often available. For instance, the physical system (computer model output) may be known to satisfy inequality constraints with respect to some or all inputs. We develop a finite-dimensional approximation of GPs capable of incorporating inequality constraints and observations (with and without noisy) for computer model emulators. It is based on a linear combination between Gaussian random coefficients and deterministic basis functions. By this methodology, the inequality constraints are respected in the entire domain. The mean and the maximum of the posterior distribution are well defined. Convergence of the method is proved and the connection with the spline estimates is done in the case of free-noisy data. This can be seen as an generalization of the Kimeldorf-Wahba (Kimeldorf and Wahba, 1971) correspondence between Bayesian estimation on stochastic process and smoothing by splines. A simulation study to show the efficiency and the performance of the proposed model in term of predictive accuracy and uncertainty quantification is included. Finally, a real application in insurance and finance to estimate a term-structure function and default probabilities is shown.