[School of Natural Sciences PhD Scholarships] Large-data limits in Markov chain Monte Carlo

About the project

About the Project We typically employ Bayesian inference in cases where, in addition to parameter inference, we are interested in quantifying the uncertainties that remain in the parameter after the data observation. This uncertainty is represented by the so-called posterior distribution. Posterior distributions are popularly approximated by Markov chain Monte Carlo (MCMC) methods or, especially in Machine Learning, by variational methods. Bayesian methods are distinguished from frequentist methods which give a point estimator along with confidence bounds. The point estimator then can usually be shown to converge to the true, data-generating parameter as more data becomes available. Interestingly, such frequentist guarantees like consistency or approximate normality (Bernstein-von Mises) can often also be derived for posterior distributions. In this project, we will study consistency and Bernstein-von Mises theorems not for full posterior distributions, but for MCMC approximations of those posteriors. We will derive associated limit theorems, construct stable algorithms, and apply the methodology in test and real world problems. We will especially focus on semi-parametric settings that have underlying physical models, non-parametric settings that appear in machine learning, and smoothing problems in data assimilation. This project is expected to start in September 2027. Before you apply: We strongly recommend that you contact the supervisors for this project before you apply. How to apply: To be considered for this project you must complete a formal application through our online application portal. If you already have an applicant account this link will directly open an application for PhD School of Natural Sciences Scholarships . If you don’t already have an applicant account, please follow the instructions here . When applying, please specify the full title and supervisor/s of the project, details of your previous study, and names and contact details of two referees. You must also upload a Supporting Statement describing your motivation to apply to the project, your CV and transcripts of awarded and in-progress university qualifications . Please note late or incomplete applications will not be considered. Equality, diversity and inclusion are fundamental to the success of The University of Manchester and central to all our activities. A diverse research community strengthens creativity, productivity and quality, while increasing the societal and economic impact of our work. We welcome applicants from all career paths, backgrounds and sections of the community, regardless of age, disability, ethnicity, gender, gender expression, sexual orientation or transgender status. We welcome applications from candidates returning to study after a career break or experience in other roles. Flexible study arrangements may be available, including part-time study at 50%, 60% or 80%, subject to the requirements of the project and funder. Eligibility : The standard academic entry requirement for this PhD is an upper second-class (2:1) honours degree (or international equivalent) in Mathematics or Statistics OR any upper-second class (2:1) honours degree and a Master’s degree at merit (or international equivalent) in Mathematics or Statistics. This project will remain open until filled. If your application is submitted by 1st November 2026, you can expect a decision by 18th December 2026. If your application is submitted by 15th January 2027, you can expect a decision by 30th March 2027. Self or externally funded students can also be considered for this project. FSESoNS

Research areas