Fast Updating of Bayesian Models

贝叶斯推理为复杂数据建模和量化不确定性提供了强大的工具，并已被用于解决许多领域的问题，包括流行病学、气候学、信号处理等。在贝叶斯推理的应用中，一个常见的要求是能够快速更新贝叶斯模型。这是特别需要的，例如，迭代地建立模型以反映先前信念的变化，将新的观察结果合并到模型中，或者在修正旧数据后纠正模型。一个具体的例子是在COVID-19传播的流行病学建模中，建立贝叶斯模型来预测每天的冠状病毒病例，每天收集新数据并将其输入模型，需要进行新的后验拟合。另一个例子是模型开发，从业者通常希望对模型的元素进行调整，以解释他们先前的信念的变化。在这种情况下，我们称之为迭代贝叶斯建模，从业者通常有一个后验样本的前一个模型在他们的处置，并希望获得一个新的样本更新的模型。传统的迭代贝叶斯建模方法是在每次模型更新时从头开始运行贝叶斯推理方法。虽然这可以产生最先进的近似精度，但这是不可取的，因为这些贝叶斯推理方法通常计算成本很高，并且用于拟合旧模型的计算将被浪费。现有文献已经提供了关于何时重新使用旧拟合可以帮助拟合新模型的理论结果，但这仍然是一个不统一的研究领域。在这个博士项目中，我们的目标是探索不同的方法，可以利用旧模型的t来加速更新模型的拟合。该项目的一些目标是：1。设计能够在假设贝叶斯模型变化形式的情况下快速更新贝叶斯模型的算法。回顾和扩展现有的贝叶斯推理方法，以允许重用以前的拟合，并对其优点和局限性进行全面比较。统一关于何时重用以前拟合的理论结果的现有文献可能比从头开始运行贝叶斯方法更有益，从而为在迭代贝叶斯建模中使用这些算法提供理论依据。该项目属于EPSRC数学科学研究领域。如果成功的话，它可以为允许快速模型更新或模型开发的编程软件的开发提供启示，从而使广泛的贝叶斯统计实践者社区受益

Bayesian inference provides a powerful tool for modelling complex data and quantifying uncertainty, and has been used to solve problems in many areas including epidemiology, climatology, signal process-ing, to name a few. In the applications of Bayesian inference, a common requirement is the ability to update Bayesian models quickly. This is especially desired when, for example, building models iteratively to reflect a change in the prior beliefs, to incorporate new observations into the model, or to correct the model after rectication of old data. A concrete example is in epidemiological modelling of the COVID-19 transmission, where a Bayesian model is built to predict the daily coronavirus cases, and new posterior fits are required every day as new data are collected and fed into the model. Another example is model development, where practitioners often wish to make adjustments to elements of themodel to account for changes in their prior believes.In this setting, which we term iterative Bayesian modelling, practitioners often have had a posterior sample for the t of the previous model at their disposal, and wish to obtain a new sample for the updated model. The traditional approach to iterative Bayesian modelling is to run a Bayesian inference method from scratch each time the model is updated. Although this can yield state-of-the-art approximation accuracy, it is undesirable since these Bayesian inference methods are often computationally expensive, and the computation spent in fitting the old models would be wasted. Existing literature has provided theoretical results on when re-using old fits could help the fitting of a new model, but this remains an ununified area of research. In this PhD project, we aim to explore different methods that can make use of the t to the old model to accelerate the fitting of the updatedmodel. Some objectives of this project are:1. Designing algorithms that are able to update Bayesian models quickly under assumptions on the form of the changes to the model.2. Reviewing and extending existing Bayesian inference methods to allow re-use of previous fits, and making comprehensive comparisons on their advantages and limitations.3. Unifying the existing literature on the theoretical results of when re-using previous fits can be more beneficial than running a Bayesian method from scratch, thus providing theoretical justificationsto the use of these algorithms in iterative Bayesian modelling.This project falls within the EPSRC research area of Mathematical Sciences. If successful, it can shed light on the development of programming software that allows rapid model updating or modeldevelopment, thereby beneting the wide community of practitioners of Bayesian statistics.1