  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 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 point
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 configuration 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 on a data set 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.
