Particle filters and extensions

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.