Flexible Tails for Normalising Flows

Principled analysis of data for decision making, whether human led or automated, requires dealing with uncertainty. The mature field of probability theory provides a wealth of theoretical support for dealing with problems of this nature. When probability is used to describe and reason about a system, the specific assumptions are referred to as probabilistic models. The central aim of probabilistic modelling is to successfully infer how the uncertainty about quantities of interest is distributed, known as probability distributions. This task is critical in many application domains that exhibit randomness, such as machine learning applications like facial recognition or the estimation of extreme values like predicting when the next flood may occur. The specification of probability distributions that are both flexible and convenient to estimate has clear value in these fields.

Normalising flows (NFs) are one such mechanism for expressing probability distributions that have generated much recent research interest. Rather than standard parametric assumptions, the central idea of NFs is to represent a potentially complex target distribution as the transformation of a simpler base distribution. The structure of the resulting distribution is controlled by both the randomness expressed by the base distribution and the form of the transformation. By requiring that the transformation is smooth and invertible, one can evaluate the NF density exactly by the well known change of variable formula in addition to efficiently drawing samples from the NF.

Recently, NF models have been successfully applied to challenging learning tasks. These include application to generating realistic images of faces and the modelling of complex physical processes. Despite the successes of NFs when performing density estimation directly, significant issues remain when applied to Bayesian inference, which commonly occurs when the quantity of interest is unobserved.

One known issue with NFs is their inability to represent distributions with heavy tails. A current solution to this problem introduces a heavy tailed base distribution. In our work, we instead propose capturing heavy tailed distributions via a transformation of uniform base distributions. Our proposal has clear technical advantages over current approaches and can rely on existing extreme value literature for theoretical support.

This direction of research has the potential to open up entirely new applications of NF models to important extreme value problems. Additionally, we will investigate whether sufficiently flexible tails could also improve Bayesian inference with NFs. Such an improvement would deliver wide ranging benefit to many important applications.

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.