Particle Filtering for Continuous Time Problems

Monte Carlo methods are a critical class of computational techniques for solving problems in Bayesian inference. When data arrives in real time or sequentially, and instantaneous statistical reasoning is required, sequential Monte Carlo (SMC) methods, or particle filters, are used to estimate the posterior distribution in real time. A major challenge in particle filtering is estimating the hidden states of a stochastic system from noisy and incomplete observations, where both the state and observations evolve continuously—a problem known as ``continuous time filtering.’’ This problem presents difficulties such as irregular observation times and the intractability of transition densities, resulting in a posterior density involving an exponential function of an intractable path integral. The accuracy of numerical approximations depends on the refinement of the time discretisation. Therefore, it is essential to develop particle filter algorithms that are accurate, cost-effective, and stable as the discretisation becomes finer or so called ``in the continuous-time limit''. Additionally, in the context of coupled conditional particle filters, the variance in the computational cost of maximal coupling for discrete distributions in conditional resampling, particularly when they are close in the discretisation refinement, can become infinitely large. This thesis contributes by developing a (nearly) unbiased particle filtering algorithm for continuous-time stochastic processes with observations arising from a Cox process. The algorithm uses Poisson estimates to approximate the intractable path integral in an unbiased manner. We demonstrate how to tune these estimates to ensure that negative values are rare and, by setting them to zero, introduce a much smaller bias compared to traditional discretisation methods. For specific cases, we quantify the probability of negative estimates and show that the particle filter remains effectively unbiased. This method is applied to parameter inference in a challenging 3D single molecule tracking problem using a Born and Wolf observation model. A second key contribution is the novel coupling construction of conditional particle filters (CPFs) using ``single event’’ killing resampling. While this approach introduces bias into maximal coupling in contrast to all event killing, we show that the bias vanishes as the discretisation size approaches zero. This methodology is further applied to construct an unbiased estimator for smoothing expectations, enabling reliable confidence intervals. Finally, the novel application to a complex diffusion bridge sampling problem is explored, where careful tuning of the discretisation is required. The contributions of this thesis are detailed across several chapters. In Chapter 3, we introduce a novel debiasing particle filtering algorithm for a class of continuous-time diffusion processes observed through a Cox process, and show that the algorithm is unbiased in the continuous-time limit. In Chapter 4, we apply this debiasing approach within the particle Markov chain Monte Carlo (particle MCMC) framework to a parameter inference task in 3D single molecule tracking using a Born and Wolf observation model. Chapter 5 presents an efficient coupling construction between two CPFs via ``single event’’ killing resampling, demonstrating that the Markov kernels from single event killing converge to those of all-event killing in the continuous-time limit. In Chapter 6, we explore the use of coupled conditional particle filters (CCPFs) to construct unbiased estimators for sampling diffusion bridges.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2024

Author dc:creator

• Jin, Ruiyang

Advisors dc:contributor.advisor

• Godsill, Simon
• Singh, Sumeetpal

• Continuous Time Filtering
• Diffusion Processes
• Particle Filters
• Time Series Estimation

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/cca22f91-d92c-493a-af3f-c0292406bf06/download
• http://purl.org/NET/rdflicense/allrightsreserved

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.119737

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/386593