Neural Inference for Complex Spatial Processes

<p dir="ltr">In spatial statistics, fitting complex, non-Gaussian spatial processes to real-world data can be challenging and non-systematic because the likelihood function and underlying statistical distributions characterizing these spatial processes are intractable to compute or simulate from respectively. Generally, the first step in model fitting involves parameter inference for the given spatial process. In a frequentist framework, the gold standard for parameter estimation and uncertainty quantification requires access to the likelihood function, yet the likelihood function is often intractable or computationally intensive. If the spatial field is not fully observed, the second step in statistical inference is spatial prediction and uncertainty quantification at unobserved locations given the partially observed data. This is done via predictive distributions, the distributions of the spatial process at the unobserved locations given the partially observed data and the parameter estimate from the first step. Conditional simulation from these predictive distributions facilitates spatial prediction and uncertainty quantification through downstream products such as conditional mean fields (spatial prediction), conditional marginal distributions (univariate uncertainty quantification), and higher-dimensional products depending on the joint conditional distribution across multiple locations. </p><p dir="ltr">Standard approximation methods to address the intractable parts of this two-step procedure–likelihood evaluation and conditional simulation from predictive distributions generally involve a tradeoff between computational efficiency and loss of information regarding the likelihood and predictive distributions, requiring burdensome and nontrivial design choices. In addition, the traditional approximation methods are often only applicable to a specific spatial process or class of spatial processes. This thesis develops a pipeline for fast and accurate statistical inference for complex spatial processes via neural approximations of the likelihood and predictive distributions without the aforementioned limitations. In the first part of the thesis, we present neural likelihood, a method for learning the likelihood via a specific classification task and a calibrated convolutional neural network (CNN) classifier, for fast and accurate parameter estimation and uncertainty quantification. Neural conditional simulation (NCS), a flexible framework for conditional simulation from predictive distributions via conditional score-based diffusion models, forms the second part of the thesis. This neural inference pipeline is amortized; for a given spatial process, one can apply this pipeline to multiple partially observed spatial fields with different observed values, different observed locations, and different model parameter estimates without retraining any of the neural networks. Finally, in the third part of this thesis, we deploy this pipeline for model fitting of complex spatial processes to highly non-Gaussian real-world data. Specifically, we fit a max-stable process to annual maxima of sea surface temperatures (SST) in the Red Sea using our neural inference pipeline, demonstrating its applicability, flexibility and scalability in a real-world setting. We conclude with a discussion on the implications of this neural inference pipeline, with a particular emphasis on NCS—a highly flexible framework with broader potential in statistical inference</p>