A Divide-and-Conquer Bayesian Approach to Large-Scale Kriging

TL;DR

This paper introduces a scalable Bayesian divide-and-conquer method called Distributed Kriging (DISK) for large-scale spatial data, enabling efficient inference and prediction by combining subset posteriors.

Contribution

The paper presents a novel distributed Bayesian approach for spatial modeling that achieves near-optimal convergence rates and scalable computation across multiple machines.

Findings

DISK achieves near-optimal convergence rates in estimating spatial surfaces.

The method scales effectively to massive datasets with distributed computing.

Validation through simulations and real data demonstrates its practical effectiveness.

Abstract

We propose a three-step divide-and-conquer strategy within the Bayesian paradigm that delivers massive scalability for any spatial process model. We partition the data into a large number of subsets, apply a readily available Bayesian spatial process model on every subset, in parallel, and optimally combine the posterior distributions estimated across 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 for predicting the responses at arbitrary locations and for inference on the model parameters and spatial surface. Based on distributed Bayesian inference, our approach is called "Distributed Kriging" (DISK) and offers significant advantages in massive data applications where the full data are stored across multiple machines. We show theoretically that the Bayes $L_2$-risk of the DISK posterior distribution achieves the near optimal convergence rate in estimating the true spatial surface with various types of covariance functions, and provide upper bounds for the number of subsets as a function of the full sample size. The model-free feature of DISK is demonstrated by scaling posterior computations in spatial process models with a stationary full-rank and a nonstationary low-rank Gaussian process (GP) prior. A variety of simulations and a geostatistical analysis of the Pacific Ocean sea surface temperature data validate our theoretical results.