Fast Fully Bayesian Gaussian Processes

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Gaussian Processes are powerful non-parametric Bayesian models, and can be used in a variety of machine learning tasks including regression, classification, generative modelling and emulation. Naive implementations of Gaussian Process regression models incur cubic time and quadratic memory costs and are not conducive to parallel compute architecture. Modern compute architecture, such as Graphics Processing Units (GPUs), are well equipped to perform matrix products in floating point arithmetic which has seen a significant growth in the applicability of deep learning methods which rely entirely on simple matrix operations. This thesis aims to critically analyse similar advances in Gaussian Process implementations that only perform matrix-vector products as their main computational overhead. Recent resurgence in the use of iterative methods to compute the relevant quantities in Gaussian Process inference has led to a wide array of techniques that tame Gaussian Processes for implementation through GPU-run frameworks, however there is no unifying view on which method to use and when. This thesis provides recommendations to help combat such a problem.