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 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 (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 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.

神经贝叶斯估计器（Neural Bayes estimators）是一类以快速且无需似然的方式逼近贝叶斯估计器的神经网络。尽管它们在估计常为计算瓶颈的空间模型中颇具吸引力，但迄今为止，空间应用中的神经贝叶斯估计器仍局限于规则网格上采集的数据。这些估计器目前还依赖于一组指定的空间位置，这意味着针对新数据集需要重新训练神经网络；这使得它们在许多应用中不切实际，并阻碍了其广泛采用。

在本研究中，我们采用图神经网络（graph neural networks, GNNs）来解决从任意空间位置采集的数据中进行参数点估计这一重要问题。除了将神经贝叶斯估计扩展至非规则空间数据外，GNNs的使用还带来了显著的计算优势：由于该估计器可适用于任意配置或数量的位置及独立重复样本，因此能够分摊给定空间模型的训练成本。

我们还通过训练一个伴随的神经贝叶斯估计器来实现快速不确定性量化（uncertainty quantification），该估计器可逼近一组边际后验分位数。我们在高斯过程和极大稳定过程上验证了所提方法。最后，我们在全球海表温度数据集上展示了该方法的效果：使用单个图形处理器（graphics processing unit），仅需数分钟即可在包含2161个空间区域的数据集上估计高斯过程模型的参数，每个区域均包含数千个非规则分布的数据点。