Optimisation-centric Generalisations of Bayesian Inference

大规模黑盒统计模型在现代社会中无处不在，旨在提供一种检查复杂系统行为的方法。例如，Imbable帮助设计了此类模型，作为RAMP倡议的一部分，该倡议旨在帮助英国政府预测新冠肺炎病毒的传播。在工程中，真实世界物理现象或资产的所谓“数字孪生”通常被用来进行虚拟压力测试，并在存在外部冲击的情况下预测关键系统的行为。这些模型的一个重要问题是，我们对它们的预测和建议的不确定性的性质。与更传统的统计分析不同，基础模型往往非常复杂，不能立即解释，而且经常被错误指定。因此，在传统统计分析范式的假设下得出的不确定性量化的标准贝叶斯方法往往是不适当的。更具体地说，它们往往会导致过度自信和缺乏稳健性。为了解决这个问题，贝叶斯不确定性量化的一般形式最近已经被开发出来。与标准贝叶斯方法相比，这种方法既能保证算法的稳健性，又能减少计算负担。这使得它们非常适合基于模拟的建模场景中的应用-例如新冠肺炎建模或数字双胞胎。然而，到目前为止，它们还没有被用于这种情况，主要是在按时间顺序的问题(如在线学习、变点检测或过滤和平滑)以及贝叶斯深度学习应用(如贝叶斯神经网络或深度高斯过程)中取得成功。尽管它们前景看好，但它们的基本理论性质以及它们的计算都是未被充分探索的研究课题。在这个联谊会中，我将推进广义贝叶斯后验的理论、方法和应用，这些后验是通过优化问题隐含地定义的。虽然这些推广的贝叶斯方法显示出了巨大的前景，但如果它们要被更广泛地采用，就需要进行这类彻底的调查。作为这一点的一部分，我将调查一个基本问题，即一个人应该如何在不同的广义后来者之间做出选择。作为补充，我将针对这些后来者的特殊性质设计贝叶斯计算方法。然后，我将利用作为这项研究的一部分所取得的进展，将它们应用于传统贝叶斯方法难以处理的两类高影响问题：围绕难以处理的可能性的模型，以及基于模拟器的推理。对于这项研究计划的应用部分，我将借鉴我的项目合作伙伴的专业知识，使用广义后验数据来更好地量化“数字双胞胎”中的不确定性，以及对国家安全具有重要意义的应用-例如对新冠肺炎大流行进行建模。

Large scale black box statistical models are ubiquitous in modern society; and aimed at providing a way to examine the behaviour of complex systems. For example, Improbable has helped design such models as part of the RAMP initiative to help the UK government predict the spread of the COVID-19 virus. In engineering, so-called 'digital twins' of real-world physical phenomena or assets are commonly used to conduct virtual stress tests and predict the behaviour of critical systems in the presence of exogenous shocks.An important concern for these models is the nature of our uncertainty about their predictions and recommendations. Unlike for more traditional statistical analysis, the underlying models are often highly complex, not immediately interpretable, and often misspecified. As a consequence, standard Bayesian methods of uncertainty quantification derived under the assumptions of the traditional paradigm for statistical analysis are often inappropriate. More specifically, they often result in over-confidence and a lack of robustness. To tackle this issue, generalised forms of Bayesian uncertainty quantification have recently been developed. Such methods can ensure robustness and reduce the computational burden relative to standard Bayesian methods. This makes them ideal for applications in simulation-based modelling scenarios---such as COVID-19 modelling or digital twins. Yet, to date they have not been used in this context and primarily enjoyed success in time-ordered problems (such as on-line learning, changepoint detection, or filtering and smoothing) as well as in Bayesian Deep Learning applications (such as Bayesian neural networks or deep Gaussian Processes). In spite of their promise however, both their foundational theoretical properties as well as their computation are under-explored topics of research. In this fellowship, I will advance the theory, methodology, and application of generalised Bayesian posteriors that are defined implicitly through an optimisation problem. While such generalised Bayesian methods have shown great promise, a thorough investigation of this kind will be required if they are to be adopted more widely. As part of this, I will investigate the fundamental question of how one should choose between different generalised posteriors. Complementing this, I will devise methodology for Bayesian computation geared towards the special properties of these posteriors. I will then leverage the advances made as part of this research to apply them on two classes of high-impact problems that traditional Bayesian methods struggle with: models revolving around intractable likelihoods, and simulator-based inference. For the applied component of this research programme, I will draw on the expertise of my project partners and use generalised posteriors for better uncertainty quantification in 'digital twins', as well as applications of importance for national security---such as modelling the COVID-19 pandemic.