gaussian-process valuation

Gaussian processes (GPs) struggle to rigorously incorporate uncountably infinite-dimensional functional prior information—such as boundary conditions or global physical constraints satisfied by PDE solutions. Method: This paper proposes a unified modeling framework grounded in reproducing kernel Hilbert spaces (RKHS), establishing for the first time a rigorous equivalence between the GP conditional expectation and orthogonal projection in RKHS. This enables direct embedding of functional constraints (e.g., Dirichlet or Neumann boundary conditions) into the GP prior, bypassing conventional pseudo-point approximations. Contribution/Results: We provide theoretical guarantees on existence, uniqueness, and convergence of the constrained GP posterior. Computationally, we design a practical numerical approximation algorithm. Experiments on PDE inverse problems demonstrate substantial improvements in uncertainty quantification accuracy and posterior consistency. The framework delivers a rigorous, general, and computationally tractable paradigm for integrating domain knowledge into Bayesian modeling.

Standard Gaussian processes (GPs) in Bayesian optimization suffer from inaccurate predictions in target regions (e.g., low-function-value areas) due to their inherent stationarity assumption, which fails on non-stationary objective functions. Method: We propose relaxed Gaussian processes (reGP), a novel GP modeling framework that weakens interpolation constraints outside the region of interest while enforcing bounded mean predictions within it. reGP further introduces a target-region-weighted prediction mechanism. Contribution/Results: This is the first systematic integration of goal-oriented principles into GP kernel design. We theoretically prove that reGP—when combined with the Expected Improvement (EI) acquisition function—guarantees global convergence. Leveraging reproducing kernel Hilbert space (RKHS) analysis, reGP significantly improves predictive accuracy and convergence speed on non-stationary functions. Empirical evaluation across diverse benchmark tasks demonstrates that reGP consistently outperforms standard stationary GPs, yielding higher-quality optima with fewer iterations.

Gaussian process regression (GPR) is often treated as a black-box surrogate model, limiting its pedagogical utility and interpretability in uncertainty quantification (UQ) for beginners. Core UQ tasks—including uncertainty propagation, risk estimation, Bayesian optimization, parameter inference, and sensitivity analysis—require deeper engagement with GPR’s inherent probabilistic structure. Method: This work develops a systematic, pedagogically grounded GPR-based UQ framework that integrates UQ-specific techniques: Bayesian quadrature, active learning, and surrogate-based sensitivity analysis. It emphasizes principled covariance kernel design, Bayesian hyperparameter estimation, and reproducible implementation. Contribution/Results: The framework lowers the barrier to applying GPR in complex UQ scenarios, enhances model transparency and decision reliability, and provides a theoretically rigorous yet practically actionable paradigm for UQ education and research across engineering and scientific disciplines.

This work addresses the lack of convergence guarantees for non-Gaussian process (non-GP) surrogate models in Bayesian optimization (BO). To resolve this, we propose the first axiomatic pseudo-BO framework, formally characterizing the minimal conditions required for sequential black-box optimization to converge. Methodologically, we design a lightweight local-regression-based surrogate model coupled with a randomized prior mechanism for efficient uncertainty quantification, and integrate it with upper-confidence-bound-type acquisition strategies. Theoretically, we provide the first rigorous convergence analysis for non-GP BO. Empirically, our framework consistently outperforms state-of-the-art methods—including GP-BO, TuRBO, and ALEBO—across high-dimensional synthetic benchmarks, neural network hyperparameter tuning, and robot control tasks. Thus, it achieves both theoretical soundness and practical superiority.

Discrete quantile estimation struggles to fully characterize continuous distributions and their uncertainties. To address this, we propose the Quantile Gaussian Process (QGP) model: it treats multi-level quantile observations as noisy measurements of the underlying quantile function, integrating a Bayesian Gaussian process prior with the asymptotic distribution theory of sample quantiles to jointly fit the distribution function and quantify uncertainty. This framework is the first to embed asymptotic statistical inference into Bayesian nonparametric quantile modeling, enabling Monte Carlo posterior sampling and calibrated probabilistic prediction. In simulations and the 2023–24 U.S. CDC influenza forecasting challenge, QGP achieves significantly improved distributional approximation accuracy, faithfully recovers parameter posterior uncertainty and quantile estimation uncertainty, and establishes a novel, interpretable, and well-calibrated paradigm for continuous probabilistic forecasting.

This work addresses the inaccuracy of lower-tail predictions in Gaussian processes (GPs) under Bayesian optimization, which arises from kernel and hyperparameter choices and undermines acquisition functions such as expected improvement. Focusing on the reliability of lower-tail predictions in noiseless settings, the paper introduces a target-oriented tail calibration framework featuring two novel concepts: spatial occurrence calibration and threshold μ-calibration. This framework establishes theoretical guarantees for predictive reliability in low-threshold regions and yields a post-processing method, termed tcGP, to enhance calibration quality. Empirical evaluations on standard benchmarks demonstrate that tcGP significantly outperforms both standard and globally calibrated GPs, improving lower-tail prediction accuracy, boosting Bayesian optimization performance, and ensuring that calibrated sampling points remain dense across the design space.

This work addresses the computational bottleneck of large-scale Gaussian processes, whose O(n³) complexity hinders efficient prediction. The authors propose a conditioning strategy based on carefully constructed data contrasts that exploits the low-rank structure of covariance matrices induced by smooth kernels. Within connected domains, the full conditional distribution can be accurately approximated using only a small number of linear combinations. This approach achieves O(T r²) offline precomputation and O(1) online prediction complexity at arbitrary locations, enabling real-time responses to unseen query points. Remarkably, the method attains near-linear overall computational cost while preserving machine-precision accuracy.

This work addresses the challenges in high-dimensional Gaussian process regression, where joint inference of multiple length-scale hyperparameters is difficult and weakly informative input variables are hard to identify. The authors propose a time-step-dependent adaptive Wishart prior that operates directly on the covariance matrix rather than on individual hyperparameters. The scale matrix of this prior is dynamically constructed using the history of Markov chain Monte Carlo (MCMC) iterations, enabling efficient Bayesian learning of hyperparameters. This approach significantly enhances the model’s ability to detect irrelevant inputs, improves learning stability, and strengthens diagnostic performance. Empirical evaluations on both synthetic and real-world datasets demonstrate the effectiveness of the proposed prior.

This work addresses the scalability challenge in Gaussian process quantile regression arising from non-conjugate likelihoods by proposing an efficient inference framework based on sparse Gaussian processes. The method introduces inducing variables to model the quantile function and leverages Laplace approximation for scalable posterior inference. It further decomposes predictive uncertainty into a conditional prior variance and a posterior inducing variance, enabling a novel dual-adaptive mechanism that dynamically optimizes inducing point locations and guides sequential data acquisition. Experimental results demonstrate that the proposed approach maintains the accuracy of Laplace approximation while significantly outperforming fixed sampling strategies, offering clear advantages in both inducing point placement efficiency and model complexity control.

Standard Gaussian process regression tends to yield overconfident posterior intervals and biased decisions when inputs are subject to measurement errors. To address this issue, this work models noisy inputs as probability measures and proposes the Deterministic Projection Wasserstein ARD Gaussian Process (PWAGP). By leveraging the Wasserstein distance, the method constructs a closed-form, positive-definite, and scalable covariance function for distributional inputs. Unlike approaches relying on latent variables or Monte Carlo sampling, PWAGP avoids stochastic approximations, thereby enhancing the transparency and robustness of uncertainty quantification. The resulting framework maintains computational efficiency while significantly improving predictive reliability under input noise.