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. They are appealing to use with spatial models and data, where estimation is often a computational bottleneck. However, 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 re-trained for new data sets; this renders them impractical in many applications and impedes their widespread adoption. In this work, we employ graph neural networks to tackle the important problem of parameter estimation from data collected over arbitrary spatial locations. In addition to extending neural Bayes estimation to irregular spatial data, our architecture leads to substantial computational benefits, since the estimator can be used with any arrangement or number of locations and independent replicates, thus amortising 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 in a global sea-surface temperature application, where we estimate the parameters of a Gaussian process model in 2,161 regions, each containing thousands of irregularly-spaced data points, in just a few minutes with a single graphics processing unit.