Efficient implementation of Gaussian process priors within flexible Bayesian hierarchical models

With the advancement of modern computational methods, Bayesian inference in complex hierarchical models has become prevalent. A main challenge for these models is the inference of unknown functions, such as disease dynamics over time. To effectively capture the variability of these unknown functions while remaining mathematically tractable, Gaussian Processes (GPs) have emerged as a popular choice of priors. Their flexibility allows for the encoding of various prior beliefs about the shape and smoothness of the unknown function, which can be fine-tuned through the covariance function to incorporate specific attributes like periodicity and differentiability. Meanwhile, their mathematical tractability makes it possible to infer complex functionals like derivatives and integrals that would otherwise be intractable. While the integration of Gaussian Processes (GPs) with Bayesian hierarchical models offers a powerful toolkit for data analysis, it comes with substantial computational demands. This limitation often hinders the practical application of GPs, especially in modern data science problems involving large datasets. To address this bottleneck, this thesis focuses on developing methods that enable the efficient use of GPs within a Bayesian framework. Specifically, I introduce a one-parameter family of GPs defined by a linear differential equation, which offers an interpretable way to encode belief about the shape and behavior of the unknown function, and encompasses various GP models suitable for tasks like smoothing and detecting quasi-periodic fluctuations. To mitigate the challenging computation, I proposed finite dimensional approximations for GPs in this family using the Finite Element Method (FEM). The strategies of these FEM approximation are motivated the property of each GP to simultaneously achieve computational efficiency and approximation accuracy. Extensive numerical studies and rigorous convergence theories substantiate the efficiency and accuracy of the approximations. To determine the appropriate prior for the parameter of this GP family that incorporates domain knowledge in an interpretable and justifiable way, I further developed a novel prior elicitation method based on the concept of predictive standard deviation (PSD). The results of this research are made accessible through open-source software, empowering researchers across various fields to apply these new methods to large and complex datasets. To demonstrate their practical utility, I present extensive applications of the proposed methods.In the context of COVID-19 risk modeling, I apply the Integrated Wiener Process from this one-parameter family to make model-based inference jointly on the function and its derivatives . This newly developed smoothing method is shown to provide efficient, accurate, and interpretable estimates of both the death rate and its rate of change. To make model-based inference of quasi-periodic functions, another class of GP from this family called seasonal Gaussian Process is introduced with applications to various analyses, including excess mortality in Canada due to COVID-19, lynx counts, sunspots, and CO2 concentration over time.

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