Bayesian Treed Vecchia Gaussian Process Emulation via Sequential Monte Carlo

Abstract

Computer models (or simulators) are used in many scientific domains to understand and predict real-world processes, but they are expensive to run. One way to address this hurdle is to use Gaussian processes (GPs). For complex physical systems such as storm surge, simulator outputs often exhibit heterogeneous structures, and many nonstationary GP methods become impractical with several inputs and tens of thousands of runs. To address these issues, we combine the divide-and-conquer idea of treed GP models with Vecchia approximations in a Bayesian treed Vecchia GP (TVGP) model. The model uses the Bayesian classification and regression tree prior for random recursive partitioning of the input domain. Within each leaf, a Vecchia approximation allows efficient likelihood evaluation. For Bayesian inference, we develop a sequential Monte Carlo algorithm that explores tree space as an alternative to reversible-jump Markov chain Monte Carlo. The method is demonstrated on synthetic examples with two and ten inputs and on a storm-surge simulator with seven inputs and up to 50,649 training runs. TVGP has the smallest prediction error in both synthetic examples and with the larger storm-surge training set, although its predictive intervals undercover in the example with ten inputs and the storm-surge application.