Using Score-based Methods for Unnormalisable Probability Density Estimation

Classical statistical modelling such as maximum likelihood estimation relies on knowledge of the normalising constant of a probability density model. Under certain cases, for example where data are observed on a generic truncated domain, the normalising constant is intractable. Whilst conventional methods usually approximate this term via numerical integration, methods such as score matching (Hyvärinen, 2005) and minimum Stein discrepancy estimators (Barp et al., 2019; Liu et al., 2016) bypass its evaluation entirely.

In chapter 3, we present a method that uses score matching to define a density estimator on a generic truncation domain. This method relies on a weighting function that is equal to zero at the boundary of the domain, for which we propose a distance function which arises naturally from maximising a Stein discrepancy. Numerical experiments show the potential of the proposed method across a range of experiments, including real world Chicago crime data, and correcting the over-compensation from outlier trimming.

In chapter 4, we extend the work from chapter 3 where data are both truncated and lie on the surface of a manifold. We present an application of this method to the sphere, and propose two distance functions as the corresponding weighting function. Experiments on the sphere show that this method achieves lower estimation error than prior Euclidean domain methods. We present a real-world application estimating the mean of storm locations truncated over the continental United States.

In chapter 5, we define an approximate Stein class for which the Stein identity holds approximately. This enables construction of a truncated kernelised Stein discrepancy, which only requires access to a set of boundary points. In experiments, we show that even with this relaxed set of assumptions, the method is comparable in performance to previous methods.

Finally, in chapter 6, we derive an estimator of the time-varying derivative of the parameter of an unnormalised exponential family density using time score matching (Choi et al., 2022). We present a changepoint detection method based on the asymptotic variance of this estimator which needs no prior assumptions on the type of changepoint expected. This highly flexible method provides comparable results to previous popular changepoint detection methods. We show its applicability to real-world applications, as well as the ability to detect when out-of distribution images are introduced to a dataset using a neural network representation.

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.