Abstract
Markovian systems are ubiquitous in nature, science, and engineering, to model the evolution of a system for which the future state of the system only depends on the past through the present state. These often appear as time series or stochastic processes, and when they are partially observed, they are known under the umbrella term of state-space models. Inferring the current state of the system from these partial, and often noisy, observations is a fundamental question in statistics and machine learning, and it is often solved using Bayesian inference methods that correct a prior belief on the state of the system through the likelihood of the observations. This perspective gives rise to typically recursive algorithms, which sequentially process the observations to slowly refine the estimate of the current state of the system. The most common of these algorithms are the Kalman filter and its extensions via linearisation procedures, and particle filtering methods, based on Monte Carlo. Another question, which often arises is that of the past state or past trajectory of the system, given all the observations. Furthermore, it may also be of interest to identify the model itself, whereby the most likely (or any other metric) model within a family is picked given the observations. In this thesis, we examine the three problems of Bayesian filtering, smoothing, and identification in the context of Markovian models, and we propose computationally efficient algorithms to solve them. In particular, we develop the parallelisation of the recursive structure of the filteringsmoothing algorithms, which, while optimal in a sequential setting, can be significantly sped up by using modern parallel computing architectures. This endeavour is tackled in both the context of particle approximations and Kalman-related methods. Another important aspect of the thesis is the use of gradient-based methods to perform inference in state-space models, taking several forms. One of these is the generalisation of the Metropolis-adjusted Langevin algorithm (MALA) and related algorithms to the context of particle and Kalman filters, and their implication for high-dimensional state inference. Another one is making particle filters differentiable by approximating the usual algorithm and then using the approximation to perform inference in statespace models using gradient-based methods. Finally, we also discuss the use of gradient-flows to perform automatic locally optimal filtering in state-space models. Some of these algorithms are de facto sequential and hardly parallelisable, but some instances can benefit from parallelisation, and we discuss the implications of this in terms of computational efficiency.
Translated title of the contribution | Computationally efficient statistical inference in Markovian models |
---|---|
Original language | English |
Qualification | Doctor's degree |
Awarding Institution |
|
Supervisors/Advisors |
|
Publisher | |
Print ISBNs | 978-952-64-1900-8 |
Electronic ISBNs | 978-952-64-1901-5 |
Publication status | Published - 2024 |
MoE publication type | G5 Doctoral dissertation (article) |
Keywords
- State-space models
- Markov chain Monte Carlo
- sequential Monte Carlo
- particle filtering
- variational inference
- parallel computing