Yonekura, Shouto;
(2020)
Some contributions to model selection and statistical inference in Markovian models.
Doctoral thesis (Ph.D), UCL (University College London).
Preview |
Text
Thesis_SY_final.pdf - Accepted Version Download (3MB) | Preview |
Abstract
The general theme of this thesis is providing and studying a new understanding of some statistical models and computational methods based on a Markov process/chain. Section 1-4 are devoted to reviewing the literature for the sake of completeness and the better understanding of Section 5-7 that are our original studies. Section 1 is devoted to understanding a Markov process since continuous and discrete types of a Markov process are hinges of the thesis. In particular, we will study some basics/advanced results of Markov chains and Ito diffusions. Ergodic properties of these processes are also documented. In Section 2 we first study the Metropolis-Hastings algorithm since this is basic of other MCMC methods. We then study more advanced methods such as Reversible Jump MCMC, Metropolis-adjusted Langevin algorithm, pseudo marginal MCMC and Hamiltonian Monte Carlo. These MCMC methods will appear in Section 3, 4 and 7. In Section 3 we consider another type of Monte Carlo method called sequential Monte Carlo (SMC). Unlike MCMC methods, SMC methods often give us on-line ways to approximate intractable objects. Therefore, these methods are particularly useful when one needs to play around with models with scalable computational costs. Some mathematical analysis of SMC also can be found. These SMC methods will appear in Section 4, 5, 6 and 7. In Section 4 we first discuss hidden Markov models (HMMs) since all statistical models that we consider in the thesis can be treated as HMMs or their generalisation. Since, in general, HMMs involve intractable objects, we then study approximation ways for them based on SMC methods. Statistical inference for HMMs is also considered. These topics will appear in Section 5, 6 and 7. Section 5 is largely based on a submitted paper titled Asymptotic Analysis of Model Selection Criteria for General Hidden Markov Models with Alexandros Beskos and Sumeetpal Sidhu Singh, https: //arxiv.org/abs/1811.11834v3. In this section, we study the asymptotic behaviour of some information criteria in the context of hidden Markov models, or state space models. In particular, we prove the strong consistency of BIC and evidence for general HMMs. Section 6 is largely based on a submitted paper titled Online Smoothing for Diffusion Processes Observed with Noise with Alexandros Beskos, https://arxiv.org/abs/2003.12247. In this section, we develop sequential Monte Carlo methods to estimate parameters of (jump) diffusion models. Section 7 is largely based on an ongoing paper titled Adaptive Bayesian Model Selection for Diffusion Models with Alexandros Beskos. In this section, we develop adaptive computational ways, based on sequential Monte Carlo samplers and Hamiltonian Monte Carlo on a functional space, for Bayesian model selection.
Type: | Thesis (Doctoral) |
---|---|
Qualification: | Ph.D |
Title: | Some contributions to model selection and statistical inference in Markovian models |
Event: | UCL |
Open access status: | An open access version is available from UCL Discovery |
Language: | English |
Additional information: | Copyright © The Author 2020. Original content in this thesis is licensed under the terms of the Creative Commons Attribution 4.0 International (CC BY 4.0) Licence (https://creativecommons.org/licenses/by/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
Keywords: | Statistics |
UCL classification: | UCL UCL > Provost and Vice Provost Offices UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Statistical Science |
URI: | https://discovery.ucl.ac.uk/id/eprint/10109466 |
Archive Staff Only
View Item |