This thesis explores the application of a Bayesian approach to hyperparameter optimization in surrogate modeling for geotechnical engineering problems. Surrogate modeling, particularly employing Gaussian Processes and Kriging, has become an essential tool for accelerating complex numerical simulations in geotechnical engineering. Traditional Maximum Likelihood Estimation (MLE) approaches to hyperparameter optimization, although straightforward, often overlook the inherent uncertainties in model hyperparameters, potentially leading to sub-optimal prediction accuracy and poor generalization.

The research presented in this thesis investigates the feasibility and benefits of applying a Bayesian inference approach to the tuning of hyperparameters in surrogate models. This approach allows for a probabilistic treatment of hyperparameters, providing a comprehensive quantification of uncertainty. The Markov Chain Monte Carlo sampling method, specifically the No-U-Turn Sampler, was employed to sample from the posterior distributions of hyperparameters, addressing the challenges posed by their non-Gaussian nature and non-linear relationship with model outputs.

Three case studies of varying complexity from geotechnical engineering practice are examined to compare the Bayesian approach against traditional MLE in terms of hyperparameter determination, prediction accuracy, uncertainty quantification, and computational efficiency. The findings suggest that the Bayesian approach, while computationally more intensive, could potentially offer more accurate predictions in terms of the mean-squared error and provide a deeper understanding of uncertainty, which is crucial for risk-informed decision-making in geotechnical engineering.

The study concludes that Bayesian hyperparameter optimization in surrogate modelling holds significant potential for improving the robustness and reliability of predictions in geotechnical engineering, particularly in applications involving complex dependencies and where a thorough understanding of uncertainty is crucial. Further research is recommended to enhance the computational efficiency of the Bayesian method and to explore its integration with multi-point enrichment strategies for practical engineering applications.