Particle filters and extensions

状态空间模型是一类广泛应用于信号处理、物理、贝叶斯推理、经济学等领域的描述系统随时间变化的统计模型。系统的状态被建模为马尔可夫过程，对于马尔可夫过程，只有部分的和有噪声的信息可以通过一些观测过程获得，目标通常是在给定可观测的情况下近似当前状态的分布--所谓的“滤波分布”。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获得。粒子过滤器是在这些设置中(以及在其他设置中)使用的一类非常流行的算法，在某些条件下能够按顺序近似过滤分布序列。该算法的性能关键取决于所使用的特定过滤器--不幸的是，理论上最有效的那些在实践中并不总是可实现的。这个项目从数学和计算统计到应用概率，将研究一些扩展粒子过滤器适用性的方法-例如，通过在算法中考虑基于马尔可夫链蒙特卡罗的步骤。该项目与EPSRC 2022-25年期间战略交付计划相一致，旨在通过研究这些计划和实际计划的趋同，从理论和数学角度研究这些问题和方法。

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.