Emerging Geometries for Statistical Science: Articulating the Vision

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.

As new theory and methods are developed for application in generic problems of statistical science, there will be diverse benefits and beneficiaries, cognate communities being built up and statisticians-at-large informed to maximise impact. Impact on a range of Academic Beneficiaries (AB) being described in the eponymous summary, we focus on others here.

NB: ICS abbreviates 'Invariant Coordinate Selection', and CIG 'Computational Information Geometry'.

IS.1 Industry
The same theory-practice iterative procedure of science outlined by Box (see AB.3) describes many instances of product development, a working model linking features of product design to product performance. Product development requiring quality control, the multivariate outlier detection capabilities of ICS offer further benefit. Industrial applications within the Thales group will be a particular focus, in collaboration with Ampère medallist Frédéric Barbaresco.

IS.2 Government
As in any time-limited decision context, policy-makers have to come to conclusions based on the data available, good practice dictating that due allowance be made for its sampling variability. Despite recent progress, what is not yet the norm is to fully allow for the effects of model uncertainty, which can be appreciable. Box again writes: 'while [diagnostic] checks are always necessary, they may not be sufficient, because some discrepancies may on the one hand be potentially disastrous and on the other be not easily detectable'. Again, even if detectable, important discrepancies may not be foreseen. The interpretability of the space of all empirically-supported important model perturbations provided by CIG gives it the potential to become a powerful tool for the policy-maker. In particular, it can help inform which contingencies policies most need to be robust against. This is especially important in safety-critical contexts. Policy issues in public health will be a particular focus, in collaboration with Bradford Hill medallist Paddy Farrington.

IS.3 Articulating the Vision
The main modes of engagement ensuring that the beneficiaries can access the potential of this research are by publication in leading journals, continued dialogue with cognate communities, and presentation at major conferences in statistics and related areas. Each mode has a successful track record, gathering momentum and well-targeted future plans, as detailed in the Case for Support. In addition to developing the WOGAS series of workshops, and initiating a complementary ICS series, events for cognate specialists will be offered at major international events, such as GSI'13.
A further way it can be ensured that beneficiaries can access the outcomes of this research will be through availability of relevant software. The development of commercial quality software is premature at present, but distribution of research quality codes will certainly form part of the standard publication process, enabling and encouraging early adoption of the project's new methodologies.
Communication of research results at international meetings will be undertaken by the PDRAs, supported by the PI. This is seen as an outstanding way for them to be trained and to establish themselves in the Statistics Community, building their careers in Statistical Methodology.

IS.4 Summary
1. Cognate research communities will be informed and 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 two PDRAs will be ideal recruits to corresponding areas of science, industry or government, as well as to higher posts in academia.