Design and applications of Monte Carlo methods based on piecewise deterministic Markov processes

Markov Chain Monte Carlo methods are the most popular algorithms used for exact Bayesian inference problems. These methods consist of simulating a Markov chain which converges to a desired Bayesian posterior measure and use the simulated trajectory to approximate expectations of functionals relative to that measure. We consider Monte Carlo methods based on Piecewise deterministic Markov processes (PDMP samplers). PDMP samplers are continuous-time processes that are non-reversible by construction. Non-reversibility may improve the performance of sampling methods, both in terms of convergence to stationarity and asymptotic variance. In Chapter 1 we give a concise presentation which motivates and introduces PDMPs. Chapter 2 is about the simulation of one-dimensional diffusion bridges. The methodology proposed relies on expanding the space of diffusion bridges with a suitable truncated basis and applying the Zig-Zag sampler on the high-dimensional coefficient space. In Chapter 3 we introduce the Boomerang sampler as a new PDMP sampler which outperforms existing PDMP samplers for target measures expressed in terms of high dimensional Gaussian measures. The Boomerang sampler has elliptical deterministic dynamics which preserves Gaussian measures at barely no cost. A key application is the simulation of diffusion bridges with the method introduced in Chapter 2 as the unnormalised density is relative to a high dimensional Gaussian measure. In chapter 4, we construct a new class of efficient Monte Carlo methods based on PDMPs suitable for inference in high dimensional mixtures of continuous and atomic components. This is achieved with the fairly simple idea of endowing existing PDMP samplers with “sticky” coordinate axes and coordinate hyper-planes. Upon hitting those subspaces, an event is triggered during which the process sticks to the subspace, this way spending some time in a sub-model. Finally, Chapter 5 presents some results on the application of PDMP samplers with boundary conditions. The key motivating applications are based on the SIR model in epidemiology used for describing the spread of diseases and hard-spheres models which are of interest in statistical mechanics.