A 3.5 year PhD position is available in the Probability and Statistics Group in the Department of Mathematics at the University of Sussex.

Monte Carlo Methods such as MCMC and SMC have played a key role in the application of Bayesian methods to real-world problems. These methods are based upon simulating discrete-time Markov processes. A number of recent developments in the last few years have given rise to a new class of MCMC and SMC methods based on continuous time Markov processes. These include the rejection-free MCMC method, the Zig-Zag sampler, the Bouncy Particle Sampler and Event Chain Monte Carlo. All of these methods involve simulating a continuous-time piecewise deterministic Markov process (PDMPs) that has been designed to be ergodic with respect to a given target posterior density. These methods are, by construction non-reversible Markov processes. There is substantial evidence to suggest that non-reversible MCMC algorithms are more efficient than their reversible counterparts. Numerical simulations also support the claim that such methods can provide a highly efficient alternative to standard MCMC methods for certain classes of posterior density.

As the main objective of MCMC is to approximate expectations in high dimensions, it is crucial that any method one uses remains efficient under increasing dimensionality. The question of the scalability of MCMC methods such as Random Walk Metropolis Hastings (RWMH), Metropolis-Adjusted-Langevin Algorithm (MALA) and Hybrid Monte Carlo have been widely studied in various contexts since the seminal papers of Roberts, Gelman and Gilks. A crucial accomplishment of this programme was the derivation of the optimal scaling of the step-size as a function of dimension, along with the identification of a target acceptance rate which is optimal in high dimensions.

The aim of this project is to understand the scalability of PDMP based methods in high dimensions. In particular, by identifying appropriate high-dimensional scaling limits for this class of process we seek to understand the influence of the parameters (such as refreshment rate, reflection kernel etc) on the performance of the sampler in high dimensions. This PhD project includes components that pull on Statistics, Probability and Numerical Analysis and is at the intersection of these three disciplines. Successful candidates will have expertise in at least one of these domains and will gain experience in the others. It is also anticipated that this project will enable the student to enhance valuable programming skills (e.g. in Julia or Python).

To accomplish these aims, the student will work closely with their supervisor Dr. A. Duncan at the University of Sussex as part of an ongoing collaboration with Dr. J. Bierkens within the Delft Institute of Applied Mathematics at TU Delft.