New developments in non-reversible Markov chain Monte Carlo

The exploration performed by a Markov chain Monte Carlo (MCMC) algorithm can be likened to the exploration of some interesting terrain. Traditional MCMC is `reversible': the simplicity of this condition has facilitated the huge number of extensions and variations on the standard MCMC algorithm that are available today; however reversibility also implies that on relatively flat terrain (and in real, high-dimensional applications only one direction is `uphill', with all other directions relatively flat), an MCMC `walker' loses their sense of direction so that their path becomes erratic and the exploration slow. By contrast, non-reversible MCMC keeps a sense of direction even over flat terrain. Current non-reversible algorithms come in two main flavours: one imagines a drone flying in a straight line above the terrain and occasionally changing direction so as to keep above the higher regions; the other inverts the terrain and imagines kicking a ball along it in a random direction. Both of these methods have great potential, but also practical problems that limit their usability. Drawing on both methods, this project will create new non-reversible algorithms which are much more efficient than standard, reversible, MCMC and can be applied across a wide variety of contexts; it will also create easy-to-use software for statistical practitioners.

MCMC is used for the statistical analysis of complex data sets across a huge range of applications, from finance and fraud detection, through understanding, predicting and intervening in the spread of infectious diseases, to understanding the location of dark matter in the universe, and our work will benefit anyone analysing complex datasets in these and many other areas.

Markov chain Monte Carlo (MCMC) is the tool of choice for statistical inference across a huge range of applications, such as those mentioned in the summary. The results of the analyses are used, for example, in deciding upon interventions to prevent the spread of infectious diseases or interventions to prevent further banking fraud, or in understanding molecular pathways and designing molecules. The more iterations of the MCMC algorithm that are performed, the more accurate and reliable the final answer, but the actual number of iterations required for `sufficient' accuracy depends on the efficiency of the underlying algorithm. All applications of MCMC are limited by finite time and finite computing resources, and for many modern, complex applications using currently-available algorithms this translates into unacceptable levels of uncertainty. Our algorithms will lead to orders of magnitude increases in the efficiency of MCMC across a wide range of applications and our easy-to-use software will enable statistical practitioners to provide clear, timely evidence to facilitate policy decisions.

Our "pathways to impact" describes a range of mechanisms for maximising the impact from this grant. Most important is the development of semi-automatic software, as mentioned above. In addition, in the final year of the grant we will also hold a workshop that will mix researchers working on non-reversible MCMC with statistical practitioners who could benefit from the efficiency of our algorithms and software, and whose specific needs for efficiency gains could also inform the final work on the grant. The research will also be disseminated at conferences and in publications. Finally, by the end of the project the PDRA will be ready to take their place working in a key national-need area, having been trained in the collaborative development of statistical methodology and in publishing related articles; in the development and documentation of associated, user-friendly statistical software; and in interaction with end users, including scientists and applied statisticians.