Exploring Probability Measures with Markov Processes
type: CreativeWork
description: In many domains where mathematical modelling is applied, a deterministic description of the system at hand is insufficient, and so it is useful to model systems as being in some way stochastic. This is often achieved by modeling the state of the system as being drawn from a probability measure, which is usually given algebraically, i.e. as a formula. While this representation can be useful for deriving certain characteristics of the system, it is by now well-appreciated that many questions about stochastic systems are best-answered by looking at samples from the associated probability measure. In this thesis, we seek to develop and analyse efficient techniques for generating samples from a given probability measure, with a focus on algorithms which simulate a Markov process with the desired invariant measure. The first work presented in this thesis considers the use of Piecewise-Deterministic Markov Processes (PDMPs) for generating samples. In contrast to usual approaches, PDMPs are i) defined as continuous-time processes, and ii) are typically non-reversible with respect to their invariant measure. These distinctions pose computational and theoretical challenges for the design, analysis, and implementation of PDMP-based samplers. The key contribution of this work is to develop a transparent characterisation of how one can construct a PDMP (within the class of trajectorially-reversible processes) which admits the desired invariant measure, and to offer actionable recommendations on how these processes should be designed in practice. The second work presented in this thesis considers the task of sampling from a probability measure on a discrete space. While work in recent years has made it possible to apply sampling algorithms to probability measures with differentiable densities on continuous spaces in a reasonably generic way, samplers on discrete spaces are still largely derived on a case-by-case basis. The contention of this work is that this is not necessary, an…
abstract: In many domains where mathematical modelling is applied, a deterministic description of the system at hand is insufficient, and so it is useful to model systems as being in some way stochastic. This is often achieved by modeling the state of the system as being drawn from a probability measure, which is usually given algebraically, i.e. as a formula. While this representation can be useful for deriving certain characteristics of the system, it is by now well-appreciated that many questions about stochastic systems are best-answered by looking at samples from the associated probability measure. In this thesis, we seek to develop and analyse efficient techniques for generating samples from a given probability measure, with a focus on algorithms which simulate a Markov process with the desired invariant measure. The first work presented in this thesis considers the use of Piecewise-Deterministic Markov Processes (PDMPs) for generating samples. In contrast to usual approaches, PDMPs are i) defined as continuous-time processes, and ii) are typically non-reversible with respect to their invariant measure. These distinctions pose computational and theoretical challenges for the design, analysis, and implementation of PDMP-based samplers. The key contribution of this work is to develop a transparent characterisation of how one can construct a PDMP (within the class of trajectorially-reversible processes) which admits the desired invariant measure, and to offer actionable recommendations on how these processes should be designed in practice. The second work presented in this thesis considers the task of sampling from a probability measure on a discrete space. While work in recent years has made it possible to apply sampling algorithms to probability measures with differentiable densities on continuous spaces in a reasonably generic way, samplers on discrete spaces are still largely derived on a case-by-case basis. The contention of this work is that this is not necessary, an…
author: Power, Samuel
language: eng
keywords: Mathematical Statistics, Computational Statistics, Algorithms, Applied Probability
license: https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/4f1238ab-7128-4bc6-821a-c6ae4ad57a9b/download
license: https://www.rioxx.net/licenses/all-rights-reserved/
date published: 2020-07-13
date modified: 2026-07-22T22:24:08Z
provider: Cambridge University
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Cambridge

PropertyValue
property ID: DOI
value: 10.17863/CAM.60096