Novel Markov chain Monte Carlo methods for high-dimensional statistics.

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Markov chain Monte Carlo (MCMC) are the tools of choice to explore complex non-standard probability distributions.These algorithms have been introduced over 60 years ago, yet it remains a very active research area as we now face increasingly difficult challenges. Namely it is now expected for these algorithms to work in high-dimensional settingsand in the presence of very large datasets.The aims and objectives of this project is to address these challenges by developing novel Markov chain Monte Carlo (MCMC)algorithms which scale to high-dimensional scenarios in a data rich enviromnent. A sharp theoretical analysis of these novel MCMC schemes will also been provided and they will be demonstrated on a variety of challenging statistical applications.Much work has been recently done on the analysis of the unadjusted Langevin algorithm in scenarios where the target distributions are log-concave.However, the log-concavity assumption is very restrictive and the unadjusted Langevin algorithm introduces some undesirable bias.We will develop novel schemes which provide consistent estimates and will aim to develop a theoretical analysis that bypasses the log-concavity assumption.In particular, we plan to focus on the development of non-reversible schemes.The longer term benefits of this project are also closely linked to the RCUK Digital Economy programme. MCMC are widely used to analyze complex datasets and can be used to develop novel collaborativefiltering and topic modelling techniques for example. It is thus expected that benefits will be experienced in the medium term by the general public; e.g. the development of more powerful search engines and recommender systems, better credit card scoring techniques, improved methods for identity fraud detection etc. Many aspect of computational finance could also readily benefit from them.