Exploring Probability Measures with Markov Processes

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, and that one can in fact define quite generally-applicable algorithms which can sample efficiently from discrete probability measures. The contributions are then to propose a small collection of algorithms for this task, and verify their efficiency empirically. Building on the previous chapter’s work, our samplers are again defined in continuous time and non-reversible, each of which offer noticeable benefits in efficiency. The third work presented in this thesis concerns a theoretical study of a particular class of Markov Chain-based sampling algorithms which make use of parallel computing resources. The Markov Chains which are produced by this algorithm are mathematically equivalent to a standard Metropolis-Hastings chain, but their real-time convergence properties are affected nontrivially by the application of parallelism. The contribution of this work is to analyse the convergence behaviour of these chains, and to use the ‘optimal scaling’ framework (as developed by Roberts, Rosenthal, and others) to make recommendations concerning the tuning of such algorithms in practice. The introductory chapters provide a general overview on the task of generating samples from a probability measure, with particular focus on methods involving Markov processes. There is also an interlude on the relative benefits of i) continuous-time and ii) non-reversible Markov processes for sampling, which are intended to provide additional context for the reading of the first two works.PhD Studentship paid for by Cantab Capital Institute for the Mathematics of Informatio

Invariant Measures for Hybrid Stochastic Systems

In this paper, we seek to understand the behavior of dynamical systems that are perturbed by a parameter that changes discretely in time. If we impose certain conditions, we can study certain embedded systems within a hybrid system as time-homogeneous Markov processes. In particular, we prove the existence of invariant measures for each embedded system and relate the invariant measures for the various systems through the flow. We calculate these invariant measures explicitly in several illustrative examples.Comment: 18 pages, 7 figure

Measure concentration through non-Lipschitz observables and functional inequalities

Non-Gaussian concentration estimates are obtained for invariant probability measures of reversible Markov processes. We show that the functional inequalities approach combined with a suitable Lyapunov condition allows us to circumvent the classical Lipschitz assumption of the observables. Our method is general and covers diffusions as well as pure-jump Markov processes on unbounded spaces