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  Robust Bayesian inference through interacting particle algorithms

   Department of Mathematical Sciences

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  Dr James Foster  No more applications being accepted  Competition Funded PhD Project (UK Students Only)

About the Project

The University of Bath is inviting applications for the following PhD project commencing in October 2023.

Supervisory Team:

Lead supervisor: Dr James Foster (Department of Mathematical Sciences, University of Bath)

Co-supervisors: Prof Neill Campbell (Department of Computer Science, University of Bath) and Tom L (GCHQ, Cheltenham, subject to contract)

Overview of the Research:

The data science revolution has been largely powered by gradient-based “single particle” optimisation. Given a parameter configuration, these algorithms make incremental adjustments so that the model can better match the training data. Despite their success, single point estimates rarely provide the uncertainty quantification needed for decision making in key applications such as healthcare.

To address this, practitioners often turn to Bayesian inference, where parameters are modelled using probability distributions instead of as single points. Extracting information from these distributions is then achievable through numerical integration. However, posterior distributions are usually high-dimensional and even simple integrals, such as the posterior mean, can be computationally challenging.

Markov Chain Monte Carlo (MCMC) is widely considered the “go-to” approach for problems in Bayesian Inference. Although MCMC has a variety of strengths, it also has inherent weaknesses due to the following properties:

  1. Monte Carlo sampling has slow convergence
  2. Markov Chains evolve sequentially

Thus, MCMC algorithms can be inaccurate (property 1) and not easily compatible with high-performance parallel computing (property 2). This has motivated the development of algorithms where clouds of interacting particles are adjusted to match the target distribution, such as “Deterministic Langevin Monte Carlo” and “Stein Variational Gradient Descent”.

This project focuses on new particle methods for Bayesian inference inspired by MCMC, but with random sampling replaced by deterministic cubature formulae from the numerical analysis literature. This new methodology (entitled “Markov Chain Cubature”) also has strong links to well-known particle algorithms for Stochastic Differential Equations (SDEs).

Although Markov Chain Cubature can avoid the weaknesses of MCMC, it introduces challenges – particularly related to parallel computing. Therefore, this PhD studentship will be interdisciplinary and supervised by academics within the Mathematics and Computer Science departments at the University of Bath.

Moreover, due to its connections with real-world applications and SDE theory, the project has gained significant interest from external collaborators, including data scientists at GCHQ and Professor Terry Lyons, who leads the Oxford-based DataSig group.

The PhD will be co-funded by GCHQ (subject to contract) and the successful candidate will join the Numerical Analysis and Data Science group at the University of Bath, which offers a lively international and interdisciplinary environment.

Project keywords: Machine learning, Bayesian inference, Data science, Particle algorithms, Stochastic processes, Mathematics.

Candidate Requirements:

Applicants should hold, or expect to receive, a First Class or good Upper Second Class UK Honours degree (or the equivalent) in either Mathematics, Computer Science or a related discipline. Background in one or more of the following areas would be advantageous:

  1. Machine learning or computational statistics
  2. Stochastic processes or applied probability
  3. Scientific computing or numerical analysis
  4. Parallel or distributed computing

The candidate should have good programming skills, ideally in Python. A master’s level qualification would also be advantageous.

Non-UK applicants must meet our English language entry requirement.

Enquiries and Applications:

Informal enquiries are encouraged and should be directed to Dr James Foster on email address [Email Address Removed].

Formal applications should be submitted via the University of Bath’s online application form for a PhD in Mathematics prior to the application deadline of Sunday 22 January 2023.

More information about applying for a PhD at Bath may be found on our website.

Funding Eligibility:

To be eligible for funding, you must qualify as a Home student. The eligibility criteria for Home fee status are detailed and too complex to be summarised here in full; however, as a general guide, the following applicants will normally qualify subject to meeting residency requirements: UK and Irish nationals (living in the UK or EEA/Switzerland), those with Indefinite Leave to Remain and EU nationals with pre-settled or settled status in the UK under the EU Settlement Scheme. This is not intended to be an exhaustive list. Additional information may be found on our fee status guidance webpage, on the GOV.UK website and on the UKCISA website.

Exceptional Overseas students (e.g. with a UK Master’s Distinction or international equivalent and relevant research experience), who are interested in this project, should contact the lead supervisor in the first instance to discuss the possibility of applying for supplementary funding.

Equality, Diversity and Inclusion:

We value a diverse research environment and aim to be an inclusive university, where difference is celebrated and respected. We welcome and encourage applications from under-represented groups.

If you have circumstances that you feel we should be aware of that have affected your educational attainment, then please feel free to tell us about it in your application form. The best way to do this is a short paragraph at the end of your personal statement.

Computer Science (8) Mathematics (25)

Funding Notes

A studentship includes Home tuition fees, a stipend (£17,668 per annum, 2022/23 rate) and research/training expenses (£1,000 per annum) for up to 3.5 years. Eligibility criteria apply – see Funding Eligibility section above.


[1] An initial prototype of Markov Chain Cubature, with accompanying presentation slides can be found at
[2] Richard D. P. Grumitt, Biwei Dai and Uroš Seljak, “Deterministic Langevin Monte Carlo with Normalizing Flows for Bayesian Inference”, Advances in Neural Information Processing Systems (NeurIPS), 2022.
[3] Qiang Liu and Dilin Wang, “Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm”, Advances in Neural Information Processing Systems (NeurIPS), 2016.
[4] Terry Lyons and Nicolas Victoir, “Cubature on Wiener Space”, Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences, 2004.

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