Zhang, X. (STAT) – Predictive Generalized Variational Inference for Spatial Gaussian Process Model

Spatially dependent data are common in environmental and climate studies, where Gaussian process models are used for prediction and uncertainty quantification. Learning their covariance structure involves both statistical and computational challenges. Likelihood-based inference may be sensitive to covariance misspecification, while repeated evaluation of prediction-oriented criteria can be computationally demanding. To study these issues, we develop a predictive generalized variational inference (GVI) framework that learns a variational distribution by balancing proper-scoring-rule losses for conditional predictions with prior-based regularization. We first develop a marginal formulation for full Gaussian process models by analytically integrating out the regression coefficients and total residual variance. This formulation reduces the optimization dimension and focuses the variational optimization on the Matérn covariance parameters, with Student-(t) conditional predictive distributions arising from the marginalization. We then develop a scalable formulation based on unordered nearest-neighbor conditionals. These conditionals define the predictive loss directly, without treating their product as a joint likelihood. Because the same marginalization is not available in this formulation, the regression coefficients and variance parameter are retained in a joint variational block. We evaluate logarithmic-score and continuous ranked probability score objectives through simulations and an analysis of December 2025 temperatures from 1,218 U.S. Historical Climatology Network stations. The simulations compare different variational formulations, scoring rules, and covariance summaries using out-of-sample predictive evaluation. The real-data analysis compares likelihood-based and predictive GVI methods through cross-validation. We also examine the computational scaling of the nearest-neighbor formulation and consider extensions based on random local covariance matrices for heterogeneous spatial dependence, as well as extensions to nonlinear regression and stochastic computer model emulation with quantitative and categorical inputs. Overall, this work studies the specification and evaluation of predictive objectives, uncertainty propagation, and computational approximation within a unified variational framework.

Event Host: Xiao Zhang, Ph.D. Student, Statistical Science 

Advisor: Bruno Sansó

Zoom: https://ucsc.zoom.us/j/95501212725?pwd=WcB4SnkvhfoUFCm0Z4wUHLGB9omS05.1

Passcode: 697077