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

A State-Space Perspective on Modelling and Inference for Online Skill Rating

This paper offers a comprehensive review of the main methodologies used for skill rating in competitive sports. We advocate for a state-space model perspective, wherein players' skills are represented as time-varying, and match results serve as the sole observed quantities. The state-space model perspective facilitates the decoupling of modeling and inference, enabling a more focused approach highlighting model assumptions, while also fostering the development of general-purpose inference tools. We explore the essential steps involved in constructing a state-space model for skill rating before turning to a discussion on the three stages of inference: filtering, smoothing and parameter estimation. Throughout, we examine the computational challenges of scaling up to high-dimensional scenarios involving numerous players and matches, highlighting approximations and reductions used to address these challenges effectively. We provide concise summaries of popular methods documented in the literature, along with their inferential paradigms and introduce new approaches to skill rating inference based on sequential Monte Carlo and finite state-spaces. We close with numerical experiments demonstrating a practical workflow on real data across different sports

Extra-corporeal membrane oxygenation in the management of 2009 influenza A (H1N1) refractory respiratory failure.

Rapidly progressive acute respiratory failure attributed to 2009 H1N1 influenza A infection has been reported worldwide-3. Refractory hypoxaemia despite conventional mechanical ventilation and lung protective strategies has resulted in the use a combination of rescue therapies, such as conservative fluid management, prone positioning, inhaled nitric oxide, high frequency oscillatory ventilation and extracorporeal membrane oxygenation (ECMO)4. ECMO allows for pulmonary or cardiopulmonary support as an adjunct to respiratory and cardiac failure, minimising ventilator-associated lung injury (VALI). This permits treatment of the underlying disease process, while concurrently allowing for recovery of the acute lung injury. This case documents a previously healthy twenty-two year old Asian male patient with confirmed pandemic (H 1N1) 2009 influenza A who was successfully managed with ECMO in the setting of severe refractory hypoxaemia and progressive hypercapnia

Assessing usability of a prototype soft exoskeleton by involving people with gait impairments

Background: Gait impairment is prevalent among many growing clinical populations e.g. people with stroke, incomplete spinal cord injury (iSCI), older adults etc. Such populations may benefit from assistive devices such as exoskeletons to improve their walking ability. XoSoft (www.xosoft.eu) is a soft exoskeleton that is being developed for people with mild to moderate gait impairments to support their mobility by providing physical actuation across joints of the lower extremities. During the design and development of a device like XoSoft, it is crucial that Primary Users (PUs, e.g. patients) are involved and provide insight into their experiences and expectations regarding device usability. However, it is still not standard practice to include PUs in rigorous testing of highly technical exoskeletons. The XoSoft consortium took an iterative design approach to the development of the XoSoft prototypes. Data from usability testing with PUs are informing next iterations of the XoSoft concepts. Purpose: The purpose of this study was to assess the PU experiences of the usability of a XoSoft prototype. This study should also highlight the importance of including PUs during the development of assistive devices. Methods: Eleven participants were recruited (mean age: 73 years, mean height: 166 cm, mean mass: 65 kg). There were three categories of PUs: frail (n=5), stroke (n=1), iSCI (n=5). Participants had no cognitive impairment (Mini Mental State Examination score > 24). The prototype consisted of a leggings-style garment with Velcro straps as anchor points for actuators across the relevant joints (hip, knee, ankle). Actuation and control was provided by a modular pneumatic/sensor controlled system, which was added to the garment in modular fashion based on the PU needs. After independent donning and doffing by the participant, the garment was donned and the Velcro straps placed and secured by a researcher to ensure proper placement. Participants then performed walking tasks with active actuation followed by completion of the System Usability Scale (SUS, Brooke 1996, maximum score = 100). Results: The scores for the SUS ranged from zero to 95 with a median rating of 52.5. The median rating corresponds to an “okay” score. According to the acceptability ranges by Bangor et al. (2008), 3 participants rated the prototype as acceptable, 3 as marginal, and 5 as not acceptable. Conclusions: The scores indicate the need for improvement in the design of future XoSoft prototypes. The large variability in SUS scores indicated that the same device may be rated considerably differently by different users. This highlights the importance of including a variety of potential users of assistive devices during development. Implications: Secondary Users (SUs) such as physical therapists are also involved in the development of XoSoft, providing insight into their own needs and the needs of their patients. The therapists also play a key role in motivating their patients to participate in research and development projects. To ensure that such assistive devices can be integrated into users’ lives and practices, it is essential to have both PUs and SUs actively involved in the technical development

Climate-informed stochastic hydrological modeling: Incorporating decadal-scale variability using paleo data

A hierarchical framework for incorporating modes of climate variability into stochastic simulations of hydrological data is developed, termed the climate-informed multi-time scale stochastic (CIMSS) framework. A case study on two catchments in eastern Australia illustrates this framework. To develop an identifiable model characterizing long-term variability for the first level of the hierarchy, paleoclimate proxies, and instrumental indices describing the Interdecadal Pacific Oscillation (IPO) and the Pacific Decadal Oscillation (PDO) are analyzed. A new paleo IPO-PDO time series dating back 440 yr is produced, combining seven IPO-PDO paleo sources using an objective smoothing procedure to fit low-pass filters to individual records. The paleo data analysis indicates that wet/dry IPO-PDO states have a broad range of run lengths, with 90% between 3 and 33 yr and a mean of 15 yr. The Markov chain model, previously used to simulate oscillating wet/dry climate states, is found to underestimate the probability of wet/dry periods >5 yr, and is rejected in favor of a gamma distribution for simulating the run lengths of the wet/dry IPO-PDO states. For the second level of the hierarchy, a seasonal rainfall model is conditioned on the simulated IPO-PDO state. The model is able to replicate observed statistics such as seasonal and multiyear accumulated rainfall distributions and interannual autocorrelations. Mean seasonal rainfall in the IPO-PDO dry states is found to be 15%-28% lower than the wet state at the case study sites. In comparison, an annual lag-one autoregressive model is unable to adequately capture the observed rainfall distribution within separate IPO-PDO states. Copyright © 2011 by the American Geophysical Union.Benjamin J. Henley, Mark A. Thyer, George Kuczera and Stewart W. Frank

Climate-informed stochastic hydrological modeling: Incorporating decadal-scale variability using paleo data

A hierarchical framework for incorporating modes of climate variability into stochastic simulations of hydrological data is developed, termed the climate-informed multi-time scale stochastic (CIMSS) framework. A case study on two catchments in eastern Australia illustrates this framework. To develop an identifiable model characterizing long-term variability for the first level of the hierarchy, paleoclimate proxies, and instrumental indices describing the Interdecadal Pacific Oscillation (IPO) and the Pacific Decadal Oscillation (PDO) are analyzed. A new paleo IPO-PDO time series dating back 440 yr is produced, combining seven IPO-PDO paleo sources using an objective smoothing procedure to fit low-pass filters to individual records. The paleo data analysis indicates that wet/dry IPO-PDO states have a broad range of run lengths, with 90% between 3 and 33 yr and a mean of 15 yr. The Markov chain model, previously used to simulate oscillating wet/dry climate states, is found to underestimate the probability of wet/dry periods >5 yr, and is rejected in favor of a gamma distribution for simulating the run lengths of the wet/dry IPO-PDO states. For the second level of the hierarchy, a seasonal rainfall model is conditioned on the simulated IPO-PDO state. The model is able to replicate observed statistics such as seasonal and multiyear accumulated rainfall distributions and interannual autocorrelations. Mean seasonal rainfall in the IPO-PDO dry states is found to be 15%-28% lower than the wet state at the case study sites. In comparison, an annual lag-one autoregressive model is unable to adequately capture the observed rainfall distribution within separate IPO-PDO states. Copyright © 2011 by the American Geophysical Union.Benjamin J. Henley, Mark A. Thyer, George Kuczera and Stewart W. Frank

Quenching massive galaxies across cosmic time with the semi-analytic model SHARK v2.0

We introduce version 2.0 of the SHARK semi-analytic model of galaxy formation after many improvements to the physics included. The most significant being: (i) a model describing the exchange of angular momentum (AM) between the interstellar medium and stars; (ii) a new active galactic nuclei feedback model which has two modes, a quasar and a radio mode, with the radio mode tied to the jet energy production; (iii) a model tracking the development of black hole (BH) spins; (iv) more sophisticated modelling of environmental effects on satellite galaxies; and (v) automatic parameter exploration using Particle Swarm Optimisation. We focus on two timely research topics: the structural properties of galaxies and the quenching of massive galaxies. For the former, SHARK v2.0 is capable of producing a more realistic stellar size-mass relation with a plateau marking the transition from disk- to bulge-dominated galaxies, and scaling relations between specific AM and mass that agree well with observations. For the quenching of massive galaxies, SHARK v2.0 produces massive galaxies that are more quenched than the previous version, reproducing well the observed relations between star formation rate (SFR) and stellar mass, and specific SFR and BH mass at $z=0$. SHARK v2.0 produces a number density of massive-quiescent galaxies >1dex higher than the previous version, in good agreement with JWST observations at $z\le 5$; predicts a stellar mass function of passive galaxies in reasonably good agreement with observations at $0.5<z<5$; and environmental quenching to already be effective at $z=5$.Comment: Submitted for publication in MNRAS. Supplementary material with additional comparisons with observations can be found here https://clagos.com/files/Shark_v2_SupplementaryMaterial.pd

Antiviral CD8(+) T Cells Restricted by Human Leukocyte Antigen Class II Exist during Natural HIV Infection and Exhibit Clonal Expansion.

CD8(+) T cell recognition of virus-infected cells is characteristically restricted by major histocompatibility complex (MHC) class I, although rare examples of MHC class II restriction have been reported in Cd4-deficient mice and a macaque SIV vaccine trial using a recombinant cytomegalovirus vector. Here, we demonstrate the presence of human leukocyte antigen (HLA) class II-restricted CD8(+) T cell responses with antiviral properties in a small subset of HIV-infected individuals. In these individuals, T cell receptor β (TCRβ) analysis revealed that class II-restricted CD8(+) T cells underwent clonal expansion and mediated killing of HIV-infected cells. In one case, these cells comprised 12% of circulating CD8(+) T cells, and TCRα analysis revealed two distinct co-expressed TCRα chains, with only one contributing to binding of the class II HLA-peptide complex. These data indicate that class II-restricted CD8(+) T cell responses can exist in a chronic human viral infection, and may contribute to immune control