{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/383806"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/383806","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Non-Gaussian Stochastic Process Priors for Learning","abstract":"Non-Gaussian statistics naturally emerge as a fundamental concept in the study of real-world phenomena where standard Gaussian models often fall short in capturing the true variability and extreme behaviour. These characteristics are especially prevalent in fields such as finance, climate science, and signal processing, where extreme events and rare fluctuations play critical roles. A suitable generalisation of the well-known Brownian motion, which is a foundational tool in Gaussian modelling, is the family of L\\'evy processes that display varying levels of heavy-tailed, non-Gaussian behaviour while maintaining the Brownian motion as an edge parameter setting. In this work, we review the theoretical background required to study non-Gaussian behaviour in continuous-time dynamical systems and spatio-temporal models based on L\\'evy processes and their extensions. Specifically, we present novel simulation methodology for the generalised inverse-Gaussian and generalised hyperbolic processes which are important classes of L\\'evy processes that were previously intractable for simulation and use in inference. We show that these simulation algorithms enable Monte Carlo inference directly in the function space of continuous-time systems based on stochastic differential equation representations. 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A suitable generalisation of the well-known Brownian motion, which is a foundational tool in Gaussian modelling, is the family of L\\'evy processes that display varying levels of heavy-tailed, non-Gaussian behaviour while maintaining the Brownian motion as an edge parameter setting. In this work, we review the theoretical background required to study non-Gaussian behaviour in continuous-time dynamical systems and spatio-temporal models based on L\\'evy processes and their extensions. Specifically, we present novel simulation methodology for the generalised inverse-Gaussian and generalised hyperbolic processes which are important classes of L\\'evy processes that were previously intractable for simulation and use in inference. We show that these simulation algorithms enable Monte Carlo inference directly in the function space of continuous-time systems based on stochastic differential equation representations. 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