Estimating Probabilistic Models on Curved Surfaces using Score Matching

The main objectives of the PhD project are to develop methodologies for probability density estimation on manifolds, particularly truncated manifolds. Statistical parameter estimation on manifolds is an underdeveloped field, and this work will have many different applications and uses across different disciplines. In many cases, it is natural to assume that some observed location-based data lie on a manifold (such as the Earth). Geographical data will benefit from a non-Euclidean approximation in measurements, which will enable more accurate estimation of key parameters such as the epicenter of a large earthquake event, or the most frequent point at which storms occur within a large country with complex boundaries such as the United States. In this example, storms will only be observed within the country borders of the United States, but many storms that lie outside of these boundaries will be un-observed, so it is important to account for this truncation effect. Furthermore, the large area of the United States would be hard to approximate as 'flat', so the region would need to be considered a curved surface as part of a manifold. In cases such as these, current estimation methods may not be as accurate due to not accounting for a spherical Earth or measurements being truncated outside the region. Adjusting the statistical model to account for truncation and manifold surfaces can be problematic due to the intractability of the normalising constant in probability density models in this case. The 'score matching' estimation method is therefore a natural candidate for extension in this regard, since it does not require the calculation this normalisation term. The key aims of the project are 1) extend the current 'score matching' methodology to density estimation on a truncated manifold by incorporating a scaling function that vanishes at the boundary, 2) develop a theoretical framework for this scaling function that controls the behaviour of the estimation method related to the manifold boundary, and 3) investigate the theoretical properties (such as consistency and bias) of the proposed estimation method. Once these primary steps are completed, the research can extend beyond the score matching method, and alternative estimation methods can be compared and contrasted with score matching to find the optimal approach to estimating probability densities on manifolds with a boundary.

The COMPASS Centre for Doctoral Training will have the following impact.

Doctoral Students Impact.

I1. Recruit and train over 55 students and provide them with a broad and comprehensive education in contemporary Computational Statistics & Data Science, leading to the award of a PhD. The training environment will be built around a set of multilevel cohorts: a variety of group sizes, within and across year cohort activities, within and across disciplinary boundaries with internal and external partners, where statistics and computation are the common focus, but remaining sensitive to disciplinary needs. Our novel doctoral training environment will powerfully impact on students, opening their eyes to not only a range of modern technical benefits and opportunities, but on the power of team-working with people from a range of backgrounds to solve the most important problems of the day. They will learn to apply their skills to achieve impact by collaborative working with internal and external partners, such as via our Rapid Response Teams, Policy Workshops & Statistical Clinics.

I2. As well as advanced training in computational statistics and data science, our students will be impacted by exposure to, and training in, important cognate topics such as ethics, responsible innovation, equality, diversity and inclusion, policy, effective communication and dissemination, enterprise, impact and consultancy skills. It is vital for our students to understand that their training will enable them to have a powerful impact on the wider world, so, e.g., AI algorithms they develop should not be discriminatory, and statistical methodologies should be reproducible, and statistical results accurately and comprehensibly communicated to the general public and policymakers.

I3. The students will gain experience via collaborations with academic partners within the University in cognate disciplines, and a wide range of external industrial & government partners. The students will be impacted by the structured training programmes of the UK Academy of Postgraduate Training in Statistics, the Bristol Doctoral College, the Jean Golding Institute, the Alan Turing Institute and the Heilbronn Institute for Mathematical Sciences, which will be integrated into our programme.

I4. Having received an excellent training, the students will then impact powerfully on the world in their future fruitful careers, spreading excellence.

Impact on our Partners & ourselves.

I5. Direct impacts will be achieved by students engaging with, and working on projects with, our academic partners, with discipline-specific problems arising in engineering, education, medicine, economics, earth sciences, life sciences and geographical sciences, and our external partners Adarga, the Atomic Weapons Establishment, CheckRisk, EDF, GCHQ, GSK, the Office for National Statistics, Sciex, Shell UK, Trainline and the UK Space Agency. The students will demonstrate a wide range of innovation with these partners, will attract engagement from new partners, and often provide attractive future employment matches for students and partners alike.

Wider Societal Impact

I6. COMPASS will greatly benefit the UK by providing over 55 highly trained PhD graduates in an area that is known to be suffering from extreme, well-known, shortages in the people pipeline nationally. COMPASS CDT graduates will be equipped for jobs in sectors of high economic value and national priority, including data science, analytics, pharmaceuticals, security, energy, communications, government, and indeed all research labs that deal with data. Through their training, they will enable these organisations to make well-informed and statistically principled decisions that will allow them to maximise their international competitiveness and contribution to societal well-being. COMPASS will also impact positively on the wider student community, both now and sustainably into the future.