A0924
Title: Vecchia Gaussian processes: Probabilistic properties and Bayesian nonparametrics
Authors: Yichen Zhu - Bocconi University (Italy) [presenting]
Botond Szabo - Bocconi University (Italy)
Abstract: Gaussian processes are widely used to model spatial dependency in geostatistical data, yet the exact computation suffers an intractable time complexity of $O(n^3)$. Vecchia approximation has become a popular solution to this computational issue, where spatial dependency is characterized by a sparse directed acyclic graph (DAG) that allows scalable Bayesian inference. Despite the popularity in practice, little is understood about the Vecchia Gaussian processes themselves, let alone their theoretical guarantees when employed in a regression model. The probabilistic properties of Vecchia Gaussian processes are systematically studied when the mother Gaussian process is a Matern process. Under minimal regularity conditions and appropriate selection of the DAG, the Vecchia Gaussian process retains many desirable properties of the mother Gaussian process. These probabilistic properties further allow the development of Bayesian nonparametric theory for the Vecchia Gaussian process, where minimax optimality is obtained when either prior smoothness matches the posterior or in an adaptive hierarchical Bayesian setting.
