Efficient inference for spatial and spatio-temporal statistical models using basis-function and deep-learning methods

Inference in spatial and spatio-temporal models can be challenging for a variety of reasons. For example, non-Gaussianity often leads to analytically intractable integrals; we may be in a “big” data setting, whereby the number of observations renders traditional methods too computationally expensive; we may wish to make inferences over spatial supports that are different to those of our measurements; or, we may wish to use a statistical model whose likelihood function is either unavailable or computationally intractable. In this thesis, I develop several techniques that help to alleviate these challenges.

First, I develop a unifying framework and accompanying software for modelling spatial and spatio-temporal data with both point- and area-support that are big, irregularly-spaced, and non-Gaussian. This framework facilitates the modelling of large data sets through the use of spatial/spatio-temporal basis functions; it caters for arbitrary observation supports by discretising the domain into basic areal units; and it caters for non-Gaussian data by employing a spatial/spatio-temporal generalised linear mixed model.

Second, I contribute to the emerging field of neural Bayes estimation. Neural Bayes estimators are neural networks that map data to point estimates of parameters; they are approximate Bayes estimators, likelihood-free, and amortised, in the sense that, once trained with simulated data, inference from observed data is extremely fast. In this thesis, I formalise the connection between neural Bayes estimators and classical point estimation, and I propose a principled way to construct neural Bayes estimators for replicated data from general statistical models via the use of permutation-invariant neural networks. The resulting estimators may be applied to data sets with an arbitrary number of replicates, and they can be used for highly parameterised spatial dependence models. Further, I tackle the important problem of neural Bayes estimation from data collected over arbitrary spatial locations, by employing graph neural networks: the resulting estimators can be used with data collected over any countable set of spatial locations, thereby amortising the cost of training for a given spatial model. Finally, I propose a novel approach to performing rigorous uncertainty quantification in an amortised manner, by training a neural Bayes estimator to jointly approximate a set of low and high marginal posterior quantiles.

To facilitate their adoption by the broader statistical community, all of the methodological contributions are incorporated in user-friendly, comprehensively-documented, open-source software packages in the Julia and R programming languages.