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Neural Bayes estimators are neural networks that approximate Bayes estimatorsin a fast and likelihood-free manner. They are appealing to use with spatialmodels and data, where estimation is often a computational bottleneck. However,neural Bayes estimators in spatial applications have, to date, been restrictedto data collected over a regular grid. These estimators are also currentlydependent on a prescribed set of spatial locations, which means that the neuralnetwork needs to be re-trained for new data sets; this renders them impracticalin many applications and impedes their widespread adoption. In this work, weemploy graph neural networks to tackle the important problem of parameterestimation from data collected over arbitrary spatial locations. In addition toextending neural Bayes estimation to irregular spatial data, our architectureleads to substantial computational benefits, since the estimator can be usedwith any arrangement or number of locations and independent replicates, thusamortising the cost of training for a given spatial model. We also facilitatefast uncertainty quantification by training an accompanying neural Bayesestimator that approximates a set of marginal posterior quantiles. Weillustrate our methodology on Gaussian and max-stable processes. Finally, weshowcase our methodology in a global sea-surface temperature application, wherewe estimate the parameters of a Gaussian process model in 2,161 regions, eachcontaining thousands of irregularly-spaced data points, in just a few minuteswith a single graphics processing unit.

Neural Bayes Estimators for Irregular Spatial Data using Graph Neural Networks