Going With the Flow: Normalizing Flows for Gaussian Process Regression under Hierarchical Shrinkage Priors

Abstract

Gaussian Process Regression (GPR) is a powerful tool for nonparametric regression, but its application in a fully Bayesian fashion in high-dimensional settings is hindered by two primary challenges: the difficulty of variable selection and the computational burden, which is particularly acute in fully Bayesian inference. This paper introduces a novel methodology that combines hierarchical global-local shrinkage priors with normalizing flows to address these challenges. The hierarchical triple gamma prior offers a principled framework for inducing sparsity in high-dimensional GPR, effectively excluding irrelevant covariates while preserving interpretability and flexibility. Normalizing flows are employed within a variational inference framework to approximate the posterior distribution of parameters, capturing complex dependencies while ensuring computational scalability. Simulation studies demonstrate the efficacy of the proposed approach, outperforming traditional maximum likelihood estimation and mean-field variational methods, particularly in high-sparsity and high-dimensional settings. This is also borne out in an application to binding affinity ($\text{pIC}_{50}$) measurements for small molecules targeting $\beta$-secretase-1 (BACE-1). The results highlight the robustness and flexibility of hierarchical shrinkage priors and the computational efficiency of normalizing flows for Bayesian GPR. This work provides a scalable and interpretable solution for high-dimensional nonparametric regression, with implications for sparse modeling and posterior approximation in broader Bayesian contexts.