Generalised Variational Bayesian Inference in Infinite Dimensions

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The advent of high-dimensional datasets, characterized by both their large size (number of observations) and feature space (the quantity of measurements per observation), has created a heightened demand for machine learning algorithms capable of handling such data. Notable successes, like ChatGPT, which harnessed a staggering 45 TB of text for training, exemplify the need for models that can handle large amounts of data.The past decade has witnessed remarkable advancements in instructing machine learning models to make precise predictions. Nonetheless, these models often lack awareness of their limitations. ChatGPT, for instance, may confidently assert erroneous information and cite fictitious articles to bolster its arguments. As a result, equipping predictive models with a measure of confidence has become a primary concern within the machine learning field known as uncertainty quantification.The literature offers numerous approaches to uncertainty quantification, many of which can be effectively summarized under the framework of generalised variational Bayesian inference. In this research endeavour, we propose employing tools from infinite-dimensional analysis, such as Gaussian measures in Banach spaces and the Wasserstein gradient flow, to scrutinize the generalised variational Bayesian inference problem. We are confident that contemporary mathematical techniques will enhance our comprehension of existing procedures and facilitate the development of new ones.Our approach will be grounded in fundamental principles, drawing from measure theory, functional analysis, and probability theory. Our primary objective is to furnish both a rigorous mathematical analysis and experimental exploration of any novel procedures we introduce. To maximize applicability, we will either construct new open-source software libraries or enhance existing ones. This fundamental research has the potential to impact a broad spectrum of applications, given the widespread adoption of machine learning algorithms and uncertainty quantification in the digital economy, engineering, scientific research, and healthcare.This project resides within the EPSRC Mathematical Sciences research domain, specifically in the field of statistics and applied probability.