Bayesian Analysis of Constrained Gaussian Processes

Bayesian analysis of constrained Gaussian processes integrates prior knowledge and observed data to model complex functions subject to known restrictions. Constrained Gaussian processes extend standard Gaussian process regression by enforcing physical, structural or logical constraints—such as monotonicity, boundedness or linear equality—directly on the latent function. This framework yields predictive distributions that rigorously respect domain requirements while providing coherent uncertainty quantification. The theoretical foundation rests on reproducing kernel Hilbert space theory, which establishes a correspondence between Gaussian process priors and optimal interpolation in functional spaces. Inference typically involves either sampling from a truncated multivariate normal posterior or optimisation to obtain maximum a posteriori estimates, the latter of which coincide with the constrained interpolant in the associated Hilbert space. Recent algorithmic advances have accelerated exact and approximate conditional simulation under high-dimensional constraints, enhanced the efficiency of sampling methods and deepened insights into the role of convex constraints within reproducing kernel frameworks. These developments have widened the applicability of constrained Gaussian processes across disciplines—from geosciences and environmental modelling to control engineering—offering robust, interpretable tools for data-driven decision-making where adherence to known constraints is crucial.

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Technical terms

Gaussian Process: A collection of random variables, any finite subset of which follows a multivariate normal distribution, used to model unknown functions.

Bayesian Inference: A probabilistic framework that updates prior beliefs with observed data to produce a posterior distribution.

Constrained Gaussian Process: A Gaussian process conditioned to satisfy specified restrictions on the latent function, such as monotonicity or boundedness.

Reproducing Kernel Hilbert Space (RKHS): A Hilbert space of functions in which evaluation at any point can be represented by an inner product with a kernel function.

Maximum A Posteriori (MAP) Estimator: The mode of the posterior distribution, often corresponding to an optimal constrained interpolant in an RKHS.

Truncated Multivariate Normal Distribution: A multivariate normal distribution restricted to a subset of its support by linear or nonlinear constraints.

Conditional Simulation: A technique for generating realisations of a random field that exactly honour observed data or constraints.

References

1. Fast Simulation of Hyperplane-Truncated Multivariate Normal Distributions. Bayesian Analysis (2017).
2. Generalization of the Kimeldorf-Wahba correspondence for constrained interpolation. Electronic Journal of Statistics (2016).
3. Conditioning geological surfaces to horizontal wells. Computational Geosciences (2022).