The long term vision of this proposed research is of statistical science enhanced by emerging geometries, driven by the needs of science, industry and government. Examples of ultimate impact include unique conspicuous benefits for experimental scientists, product development teams and policy-makers. The fundamental driver for this vision is that, given a statistical problem, an appropriate geometry can inform a novel, enhanced methodology for it. Colloquially: 'use the right tool for the job'.
Statistics, with its procedures for reasoning under uncertainty, is deeply embedded across science, industry and government. A picture being worth a thousand words, while requiring invariance to irrelevant choices, many of its methods are based on geometry.
The resulting invariant insights come at a price - that of finding a match between, on the one hand, underlying geometric axioms and, on the other, statistical conditions appropriate to a given applied context. Whereas global Euclidean geometry matches many contexts very well, increasingly, advances and challenges in science and elsewhere are throwing up important problems which demand that alternatives be used. A variety of geometries - affine, convex, differential, algebraic - have been emerging to meet these challenges.
To ensure maximal impact and provide the appropriate context in which to focus the advances to be made in theoretical and methodological development, this project targets 3 generic statistical problems where such alternative geometries are required. These problems present some of the most exacting challenges to statistical methodology while offering vast potential in application:
(1) dealing with model uncertainty,
(2) estimating mixtures and
(3) analysing high dimensional low sample size data.
Each was central to a recent cutting-edge event hosted, respectively, by the Royal Society, the International Centre for Mathematical Sciences and the Isaac Newton Institute, their identified fields of application including: theoretical physics, cosmology, biology, economics, health, image analysis, microarray analysis, finance, document classification, astronomy and atmospheric science, as well as the media, government and business.
Rooted in two new research areas - invariant coordinate selection and computational information geometry - this ambitious programme will bring together and extend emerging geometries for these important generic statistical problems. Developing the necessary underlying theory, it will provide novel, geometrically-enhanced, methodologies as tools for practical application. Pursuing potentially transformative blue sky lines of enquiry, it will enlarge both research areas leading to further new methodologies. In concert with cognate research communities, it will widely articulate the overall vision announced above.
Ultimately, this work will have a very broad impact. The following specific pathways to this end have been identified, embedded statisticians facilitating pathways 2 to 4:
1. Cognate research communities will be stimulated by advances in mathematical and computational statistics, fundamental theory underpinning new methodologies.
2. Science can ultimately benefit from more efficient theory-practice iteration.
3. The economy can ultimately benefit from faster, better product development.
4. Society can ultimately benefit from more robust policy-making.
5. With their project-enhanced transferable skills, the 2 PDRAs will be ideal recruits to many areas of science, industry or government, as well as to higher posts in academia.
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