The vision of this research is to formalise the geometric foundations of computational statistics and provide the tools and analytic results required to realise the ambition of developing the advanced statistical methodology that is essential to address emerging inference problems of major importance across the sciences and industry. As ever more demanding and ambitious applications of existing statistical inference methods are being considered, the capabilities of computational statistics tools are constantly being stretched, often beyond what is practically feasible. For example the potential to gain insights into the mechanisms of cellular function, elucidating ecological dynamics; improving neurological diagnostics, and uncovering the deep mysteries of the cosmos are only some of the ongoing scientific studies that are heavily reliant on statistical inference methods and are placing unparalleled demand on the current capabilities of available statistical methodology. This situation motivates continual innovation in the development of statistical methods for the quantification of uncertainty. The aim of this proposed research is to be more ambitious and go much further in establishing a novel paradigm that underpins the advancement of next generation computational statistical methods by formalising and developing advanced Monte Carlo methods. The geometric foundations of computational statistics will be formalised within this proposed research in a way that reaches beyond traditional interfaces between statistical and mathematical sciences.

The Oxford-Warwick Statistics Programme will train a new cohort of at least 50 graduates in the theory, methods and applications of Statistical Science for 21st Century data-intensive environments and large-scale models. This is joint project lead by the Statistics Departments of Oxford and Warwick. These two departments, ranked first and second for world leading research in the last UK research assessment exercise, can provide a wonderful stimulating training environment for doctoral students in statistics. The Centre's pool of supervisors are known for significant international research contributions in modern computational statistics and related fields, contributions recognised by over 20 major National and International Awards since 2008. Oxford and Warwick attract students with competitively won international scholarships. The programme leaders expect to expand the cohort to 11 or 12 per year by bringing these students into the CDT, and raising their funding up to CDT-level using £188K in support from industry and £150K support from donors. The need to engage in large-scale highly structured statistical models has been recognized for some time within areas like genomics and brain-imaging technologies. However, the UK's leading industries and sciences are now also increasingly aware of the enormous potential that data-driven analysis holds. These industries include the engineering, manufacturing, pharmaceutical, financial, e-commerce, life-science and entertainment sectors. The analysis bottleneck has moved from being able to collect and record relevant data to being able to interpret and exploit vast data collections. These and other businesses are critically dependent on the availability of future leaders in Statistics, able to design and develop statistical approaches that are scalable to massive data. The UK can take a world lead in this field, being a recognized international leader in Statistics; and OxWaSP is ideally placed to realize the potential of this opportunity. The Centre is focused on a new type of training for a new type of graduate statistician in statistical methodology and computation that is scalable to big data. We will bring a new focus on training for research, by teaching directly from the scientific literature. Students will be thrown straight into reading and summarizing journal papers. Lecture-format contact is used sparingly with peer-to-peer learning central to the training approach. This is teaching and learning for research by doing research. Cohort learning will be enhanced via group visits to companies, small groups reproducing results from key papers, student-orientated paper discussions, annual workshops and a three-day off-site retreat. From the second year the students will join their chosen supervisors in Warwick and Oxford, five in each Centre coming together regularly for research group meetings that overlap Oxford and Warwick, for workshops and retreats, and teaching and mentoring of students in earlier years. The Centre is timely and ambitious, designed to attract and nurture the brightest graduate statisticians, broadening their skills to meet the new challenge and allowing them to flourish in a focused, communal, research-training environment. The strategic vision is to train the next generation of statisticians who will enable the new data-intensive sciences and industries. The Centre will offer a vehicle to bring together industrial partners from across the two departments to share ideas and provide an important perspective to our students on the research challenges and opportunities within commercial and social enterprises. Student's training will be considerably enhanced through the Centre's visits, lectures, internships and co-supervision from global partners including Amazon, Google, GlaxoSmithKline, MAN and Novartis, as well as smaller entrepreneurial start-ups Deepmind and Optimor.

The aim of this network is to establish the UK as the world leading authority in the joint area of Computational Statistics and Machine Learning (CompStat & ML) by advancing communication, interchange and collaboration within the UK between the disciplines of Computational Statistics (CompStat) and Machine Learning (ML). The UK has tremendous research strength and depth that is widely acknowledged as world leading in both the individual areas of Computational Statistics and Machine Learning. Despite each of these fields of research developing, largely, independently and having their own separate journals, international societies, conferences and curricula both areas of investigation share a common theoretical foundation based on the underlying formal principles of mathematical statistics and statistical inference. As such there is a natural diffusion of concepts, research and individuals between both disciplines. This network will seek to formalise as well as enhance this interchange and in the process capitalise on important synergies that will emerge from the combined and shared research agendas of CompStat & ML.

The goal of this project is to study ways to improve the performance of large scale networks, like the Internet, in the presence of selfish entities. This can only be achieved with a better understanding of these environments and their operational points, called Nash equilibria. A Nash equilibrium is a state in which no player improve their utility by changing to another strategy. In this project we focus on the very fundamental class of congestion games and the related class of potential games. In a congestion game, we are given a set of resources and each player selects a subset of them (e.g. a path in a network). Each resource has a cost function that depends on the load induced by the players that use it. Each player aspires to minimise the sum of the resources' costs in its strategy given the strategies chosen by the other players. Such games are expressive enough to capture a number of otherwise unrelated applications - including routing and network design - yet structured enough to permit a useful theory. In this project, we will push the frontiers of this theory even further. Moreover, in collaboration with XEROX, we will investigate applications of this theory to demand management in transportation systems. Two of these applications are smart road toll and parking management systems. Congestion games have attracted lots of research, but many fundamental problems are still open. We have identified three important directions in which we want to extent the current state-of-the-art. These are: (1) evaluation of Nash equilibria (2) computational complexity of Nash equilibria (3) approximation of optimal solutions

The vision of this research is to formalise the geometric foundations of computational statistics and provide the tools and analytic results required to realise the ambition of developing the advanced statistical methodology that is essential to address emerging inference problems of major importance across the sciences and industry. As ever more demanding and ambitious applications of existing statistical inference methods are being considered, the capabilities of computational statistics tools are constantly being stretched, often beyond what is practically feasible. For example the potential to gain insights into the mechanisms of cellular function, elucidating ecological dynamics; improving neurological diagnostics, and uncovering the deep mysteries of the cosmos are only some of the ongoing scientific studies that are heavily reliant on statistical inference methods and are placing unparalleled demand on the current capabilities of available statistical methodology. This situation motivates continual innovation in the development of statistical methods for the quantification of uncertainty. The aim of this proposed research is to be more ambitious and go much further in establishing a novel paradigm that underpins the advancement of next generation computational statistical methods by formalising and developing advanced Monte Carlo methods. The geometric foundations of computational statistics will be formalised within this proposed research in a way that reaches beyond traditional interfaces between statistical and mathematical sciences.