Particle filters and extensions

状态空间模型是一种流行且灵活的统计模型，广泛用于信号处理、物理、贝叶斯推理、经济学和其他领域，用于描述随时间变化的系统。系统的状态被建模为一个马尔可夫过程，其中只有一些部分和有噪声的信息可以通过一些观察过程获得，目标通常是在给定可观测值的情况下近似当前状态的分布——即所谓的“过滤分布”。不幸的是，除了在简单的设置中，这种分布通常不能以分析形式提供。粒子滤波器是一种非常流行的算法，用于这些设置（以及其他设置），在某些条件下，能够依次近似过滤分布的序列。该算法的性能主要取决于所使用的特定过滤器——不幸的是，理论上最有效的过滤器在实践中并不总是可以实现的。这个项目涵盖了从数学和计算统计学到应用概率，将研究一些方法来扩展粒子滤波器的适用性——例如，在算法中考虑基于马尔可夫链蒙特卡罗的步骤。该项目符合EPSRC 2022-25年期间的战略交付计划，旨在通过研究这些方案的收敛性，从理论和数学的角度研究这些问题和方法，并从实践的角度进行研究。

State-space models are a popular and flexible class of statistical models widely used in signal processing, physics, Bayesian inference, economics, and other fields to describe systems which change over time. The state of the system is modelled as a Markov process for which only some partial and noisy information is available via some observation process, and the goal is often to approximate the distribution of the current state given the observables available - the so-called "filtering distributions". Unfortunately, except in simple settings, this distribution is often not available in analytical form. Particle filters are a very popular class of algorithms used in these settings (and in others as well) that - under some conditions - are able to approximate sequentially the sequence of filtering distributions. The performance of this algorithm depends crucially on the particular filter used - and unfortunately those which are theoretically the most efficient are not always implementable in practice. This project, spanning from mathematical and computational statistics to applied probability, will investigate some methods to extend the applicability of particle filters - for instance by considering Markov Chain Monte Carlo based steps within the algorithm. The project, in line with the EPSRC strategic delivery plan for the 2022-25 period, aims to study these issues and methodology from both a theoretical and mathematical point of view by studying the convergence of such schemes, and a practical one.