Efficient constrained Gaussian process approximation using elliptical slice sampling
Résumé
In this paper, Bayesian shape-restricted function estimation using constrained Gaussian processes (GPs) is revisited. The finite-dimensional Gaussian process approximation proposed in [H. Maatouk and X. Bay. Gaussian process emulators for computer experiments with inequality constraints. Mathematical Geosciences, 49(5):557–582, 2017] is considered. This approximation verifies a wide range of shape constraints such as monotonicity, convexity and boundedness constraints in the entire domain. Through this approach, shape constraints are reformulated as equivalent linear inequality constraints on the basis coefficients. To generate a sample from the resulting constrained posterior distribution, we employ a recently efficient circulant embedding technique. This technique involves absorbing a smooth relaxation of the constraint set into the likelihood, a prior distribution, and elliptical slice sampling (ESS). Our contribution in this article is threefold. First, we extend this technique to address sets of linear, quadratic and nonlinear inequalities, enabling the incorporation of more general and multiple shape constraints. These constraints can be applied individually, jointly, and sequentially. Furthermore,
this generalization allows the proposed approach to be easily adapted to other basis functions and models. Second, we explore efficient samplers to approximate both the posterior and prior distributions, including Hamiltonian Monte Carlo and the Fast Fourier Transform. Furthermore, we employ a highly efficient, large-scale approach for sampling from the prior distribution, resulting in significant computational advantages. Third, we investigate the capability of this approach to handle higher-dimensional input spaces and manage a large number of observations. The proposed approach demonstrates flexibility, accuracy, and efficiency in both synthetic and real data studies.
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