B0916
Title: Scaling of piecewise deterministic Monte Carlo for anisotropic targets
Authors: Kengo Kamatani - ISM (Japan) [presenting]
Gareth Roberts - University of Warwick (United Kingdom)
Joris Bierkens - Delft Institute of Applied Mathematics (Netherlands)
Abstract: Piecewise deterministic Markov processes (PDMPs) are a type of continuous-time Markov process that combines deterministic flows with jumps. Recently, PDMPs have garnered attention within the Monte Carlo community as a potential alternative to traditional Markov chain Monte Carlo (MCMC) methods. The Zig-Zag sampler and the Bouncy particle sampler are commonly used examples of the PDMP methodology which have also yielded impressive theoretical properties, but little is known about their robustness to extreme dependence or isotropy of the target density. It turns out that PDMPs may suffer from poor mixing due to anisotropy and this effect is investigated in detail in the stylised but important Gaussian case. To this end, a multi-scale analysis framework is employed. The results show that when the Gaussian target distribution has two scales, of order 1 and, the computational cost of the Bouncy particle sampler is of order $^(-1)$, and the computational cost of the Zig-Zag sampler is either $^(-1)$ or $^(-2)$, depending on the target distribution. In comparison, the cost of the traditional MCMC methods such as RWM or MALA is of order $^(-2)$, at least when the dimensionality of the small component is more than 1. Therefore, there is a robustness advantage to using PDMPs in this context.
