Generalised Variational Bayesian Inference in Infinite Dimensions

高维数据集的出现，其特征在于它们的大尺寸（观测数量）和特征空间（每个观测的测量数量），对能够处理这些数据的机器学习算法产生了更高的需求。值得注意的成功，如ChatGPT，它利用了惊人的45 TB的文本进行训练，满足了对能够处理大量数据的模型的需求。过去十年，在指导机器学习模型进行精确预测方面取得了显着进步。然而，这些模式往往缺乏对其局限性的认识。例如，ChatGPT可能自信地断言错误信息，并引用虚构的文章来支持其论点。因此，为预测模型配备置信度度量已成为机器学习领域的主要关注点，称为不确定性量化。文献提供了许多不确定性量化的方法，其中许多方法可以在广义变分贝叶斯推理的框架下有效地总结。在这项研究中，我们建议采用无限维分析工具，如在Banach空间和Wasserstein梯度流的高斯措施，审查广义变分贝叶斯推理问题。我们相信，当代数学技术将提高我们对现有程序的理解，并促进新程序的开发。我们的方法将以基本原理为基础，从测度论，泛函分析和概率论中提取。我们的主要目标是提供一个严格的数学分析和实验探索的任何新的程序，我们介绍。为了最大限度地提高适用性，我们将构建新的开源软件库或增强现有的软件库。鉴于机器学习算法和不确定性量化在数字经济、工程、科学研究和医疗保健中的广泛应用，该基础研究有可能影响广泛的应用。该项目属于EPSRC数学科学研究领域，特别是统计和应用概率领域。

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.