Functional Analysis of Markov Chain Monte Carlo algorithms

Successful implementation of Markov Chain Monte Carlo (MCMC) techniques relies largely on their speed of convergence to equilibrium. Many attempts have been done in order to characterise the convergence rate of a Markov Chain: when the rate of convergence is geometric, one can investigate the spectrum of the Markov operator and obtain the rate of convergence starting from its second largest eigenvalue. See e.g. Douc, R., E. Moulines, P. Priouret, and P. Soulier (2018). Of particular interest in this framework is the broad class of the so-called "simulated tempering" operators, whose spectrum can be bounded by the spectra of single chains moving within sets of the state space forming a partition, along the way of techniques for the study of Markov processes in Molecular Kinetics and Thermodynamics. See for example Madras, N. and D. Randall (2002).

Not all Markov chains converge geometrically. An algorithm's speed of convergence may slow down depending on the shape and structure of the target and proposal distributions. A typical example can be found in the Particle MCMC area, where, depending on the unboundedness of some importance weights, the Particle Gibbs Sampler may be sub-geometrically convergent. See Andrieu, C., A. Lee, and M. Vihola (2015).

Particle MCMC methods are of great interest in this project. Still quite unexplored, they arise from Physics as a Monte Carlo approximation of a discrete-time counterpart of some solution of the Schrödinger equation. See Del Moral, P. (2004). Among the major advantages of these algorithms is the fact that they provide unbiased estimates of quantities of interest. Their implementation, in practice, relies on importance resampling techniques, and hence on certain weights depending on the target and proposal distributions. These weights, as mentioned above, have very often a critical role in the performance of the algorithm, that is, may lead to sub-geometric convergence depending on their specification.

To recover the convergence rate in the sub-geometric case is still a quite unexplored area. Only recently, Andrieu, C., A. Lee, S. Power, and A. Q. Wang (2021) provided us with a novel technique based on Functional Analysis results, such as Poincaré Inequalities, which allow to determine the convergence rate of a sub-geometrically convergent Markov Chain. This opens up new opportunities of generalising many of the results from the geometric case to the sub-geometric case.

References
Douc, R., E. Moulines, P. Priouret, and P. Soulier (2018)
Madras, N. and D. Randall (2002)
Del Moral, P. (2004)
Andrieu, C., A. Lee, and M. Vihola (2015)
Andrieu, C., A. Lee, S. Power, and A. Q. Wang (2021)

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.