Neural Bayes Estimators for Irregular Spatial Data Using Graph Neural Networks

Neural Bayes estimators are neural networks that approximate Bayes estimators in a fast and likelihood-free manner. Although they are appealing to use with spatial models, where estimation is often a computational bottleneck, neural Bayes estimators in spatial applications have, to date, been restricted to data collected over a regular grid. These estimators are also currently dependent on a prescribed set of spatial locations, which means that the neural network needs to be retrained for new datasets; this renders them impractical in many applications and impedes their widespread adoption. In this work, we employ graph neural networks (GNNs) to tackle the important problem of parameter point estimation from data collected over arbitrary spatial locations. In addition to extending neural Bayes estimation to irregular spatial data, the use of GNNs leads to substantial computational benefits, since the estimator can be used with any configuration or number of locations and independent replicates, thus, amortizing the cost of training for a given spatial model. We also facilitate fast uncertainty quantification by training an accompanying neural Bayes estimator that approximates a set of marginal posterior quantiles. We illustrate our methodology on Gaussian and max-stable processes. Finally, we showcase our methodology on a dataset of global sea-surface temperature, where we estimate the parameters of a Gaussian process model in 2161 spatial regions, each containing thousands of irregularly-spaced data points, in just a few minutes with a single graphics processing unit. Supplementary materials for this article are available online.