Statistical models for spatial processes play a central role in statistical analyses of spatial data. Yet, it is the simple, interpretable, and well understood models that are routinely employed even though, as is revealed through prior and posterior predictive checks, these can poorly characterise the spatial heterogeneity in the underlying process of interest. Here, we propose a new, flexible class of spatial-process models, which we refer to as spatial Bayesian neural networks (SBNNs). An SBNN leverages the representational capacity of a Bayesian neural network; it is tailored to a spatial setting by incorporating a spatial "embedding layer" into the network and, possibly, spatially-varying network parameters. An SBNN is calibrated by matching its finite-dimensional distribution at locations on a fine gridding of space to that of a target process of interest. That process could be easy to simulate from or we have many realisations from it. We propose several variants of SBNNs, most of which are able to match the finite-dimensional distribution of the target process at the selected grid better than conventional BNNs of similar complexity. We also show that a single SBNN can be used to represent a variety of spatial processes often used in practice, such as Gaussian processes and lognormal processes. We briefly discuss the tools that could be used to make inference with SBNNs, and we conclude with a discussion of their advantages and limitations.

翻译：空间过程的统计模型在空间数据统计分析中占据核心地位。然而，即便通过先验和后验预测检验揭示出这些模型可能无法准确刻画目标过程的空间异质性，实践中仍主要采用简单、可解释且易于理解的模型。本文提出一类新型灵活的空间过程模型，称为空间贝叶斯神经网络（SBNN）。SBNN利用贝叶斯神经网络的表征能力，通过引入空间"嵌入层"并允许网络参数随空间位置变化，使其适配空间设定。通过在精细空间网格上匹配目标过程的有限维分布对SBNN进行校准，该目标过程可便于模拟或已有大量观测样本。我们提出多种SBNN变体，其中大部分在选定网格上匹配目标过程有限维分布的能力优于相同复杂度的传统贝叶斯神经网络。研究还表明，单一SBNN即可表征高斯过程、对数正态过程等实践常用的多种空间过程。本文简要讨论了可用于SBNN推断的工具，并总结了其优势与局限性。