# High-Dimensional Bayesian Geostatistics

## TL;DR

This paper reviews two scalable Bayesian spatiotemporal modeling approaches—low-rank processes and Nearest-Neighbor Gaussian Processes—that enable efficient analysis of large geostatistical datasets by reducing computational complexity.

## Contribution

It introduces and compares two novel methods for constructing scalable Bayesian spatiotemporal priors suitable for large datasets, addressing computational challenges in hierarchical models.

## Key findings

- Both methods achieve linear computational complexity in the number of locations.
- The approaches facilitate full Bayesian inference for large-scale spatiotemporal data.
- Comparison provides insights into their methodological differences and applications.

## Abstract

With the growing capabilities of Geographic Information Systems (GIS) and user-friendly software, statisticians today routinely encounter geographically referenced data containing observations from a large number of spatial locations and time points. Over the last decade, hierarchical spatiotemporal process models have become widely deployed statistical tools for researchers to better understand the complex nature of spatial and temporal variability. However, fitting hierarchical spatiotemporal models often involves expensive matrix computations with complexity increasing in cubic order for the number of spatial locations and temporal points. This renders such models unfeasible for large data sets. This article offers a focused review of two methods for constructing well-defined highly scalable spatiotemporal stochastic processes. Both these processes can be used as "priors" for spatiotemporal random fields. The first approach constructs a low-rank process operating on a lower-dimensional subspace. The second approach constructs a Nearest-Neighbor Gaussian Process (NNGP) that ensures sparse precision matrices for its finite realizations. Both processes can be exploited as a scalable prior embedded within a rich hierarchical modeling framework to deliver full Bayesian inference. These approaches can be described as model-based solutions for big spatiotemporal datasets. The models ensure that the algorithmic complexity has $\sim n$ floating point operations (flops), where $n$ the number of spatial locations (per iteration). We compare these methods and provide some insight into their methodological underpinnings.