Distributed Bayesian Inference in Massive Spatial Data

Abstract

Gaussian process (GP) regression is computationally expensive in spatial applications involving massive data. Various methods address this limitation, including a small number of Bayesian methods based on distributed computations (or the divide-and-conquer strategy). Focusing on the latter literature, we achieve three main goals. First, we develop an extensible Bayesian framework for distributed spatial GP regression that embeds many popular methods. The proposed framework has three steps that partition the entire data into many subsets, apply a readily available Bayesian spatial process model in parallel on all the subsets, and combine the posterior distributions estimated on all the subsets into a pseudo posterior distribution that conditions on the entire data. The combined pseudo posterior distribution replaces the full data posterior distribution in prediction and inference problems. Demonstrating our framework’s generality, we extend posterior computations for (nondistributed) spatial process models with a stationary full-rank and a nonstationary low-rank GP priors to the distributed setting. Second, we contrast the empirical performance of popular distributed approaches with some widely-used, nondistributed alternatives and highlight their relative advantages and shortcomings. Third, we provide theoretical support for our numerical observations and show that the Bayes ${\mathit{L}}_2$-risks of the combined posterior distributions obtained from a subclass of the divide-and-conquer methods achieves the near-optimal convergence rate in estimating the true spatial surface with various types of covariance functions. Additionally, we provide upper bounds on the number of subsets to achieve these near-optimal rates.