Dr Michael Faulkner

Piecewise deterministic Markov processes for the physical sciences

Sampling algorithms are commonplace in statistics and machine learning – in particular, in Bayesian computation – and have been used for decades to enable inference, prediction and model comparison in many different settings. They are also widely used in statistical physics, where many popular sampling algorithms first originated, e.g., the Metropolis [1] and molecular-dynamics [2] algorithms. Both algorithms have led to huge success, but the former often exhibits slow mixing (particularly in flattened regions of the probability landscape found at phase transitions) while the latter suffers from numerical instabilities (which can become significant when relaxing molecular systems in physical chemistry and biophysics). More recent developments in both Bayesian computation [3] and statistical physics [4] have led, however, to state-of-the-art sampling algorithms that drive the system through its state space with ballistic-style dynamics. These piecewise deterministic Markov processes (PDMPs) are event driven which precludes any numerical instability. They are therefore the optimal candidate for tackling both issues. This talk gives a brief introduction to statistical physics for the statistician, before moving on to Metropolis and PDMP simulations of the hard-disk model. We then present PDMPs for smooth probability models, before comparing and contrasting accelerated PDMP mixing (relative to Metropolis) of an important example from statistical physics [5]. If time allows, we will discuss the use of exact PDMP subsampling for resolving long-range interactions (i.e., significant coupling between each component of the parameter) [6]. This talk uses concepts developed in a recent review paper on the shared structure of the two research fields [7].

[1] Metropolis et al., J. Chem. Phys. 21 1087 (1953)
[2] Alder & Wainwright, J. Chem. Phys. 27 1208 (1957)
[3] Bierkens & Roberts, Ann. Appl. Probab. 27, 846 (2017)
[4] Bernard, Krauth & Wilson, Phys. Rev. E 80 056704 (2009)
[5] Faulkner, Phys. Rev. B 109, 085405 (2024)
[6] Kapfer & Krauth, Phys. Rev. E 94, 031302 (2016); Faulkner et al., J. Chem. Phys. 149, 064113 (2018)
[7] Faulkner & Livingstone, Statist. Sci. 39, 137 (2024)