We construct a new class of efficient Monte Carlo methods based on continuous-time piecewise deterministic Markov processes (PDMPs) suitable for inference in high dimensional sparse models, i.e. models for which there is prior knowledge that many coordinates are likely to be exactly 0. This is achieved with the fairly simple idea of endowing existing PDMP samplers with “sticky” coordinate axes, coordinate planes etc. 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. This results in non-reversible jumps between different (sub-)models. While we show that PDMP samplers in general can be made sticky, we mainly focus on the Zig-Zag sampler. Compared to the Gibbs sampler for variable selection, we heuristically derive favourable dependence of the Sticky Zig-Zag sampler on dimension and data size. The computational efficiency of the Sticky Zig-Zag sampler is further established through numerical experiments where both the sample size and the dimension of the parameter space are large.@en

We introduce the use of the Zig-Zag sampler to the problem of sampling conditional diffusion processes (diffusion bridges). The Zig-Zag sampler is a rejection-free sampling scheme based on a non-reversible continuous piecewise deterministic Markov process. Similar to the Lévy–Ciesielski construction of a Brownian motion, we expand the diffusion path in a truncated Faber–Schauder basis. The coefficients within the basis are sampled using a Zig-Zag sampler. A key innovation is the use of the fully local algorithm for the Zig-Zag sampler that allows to exploit the sparsity structure implied by the dependency graph of the coefficients and by the subsampling technique to reduce the complexity of the algorithm. We illustrate the performance of the proposed methods in a number of examples.@en

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.@en

This paper introduces the boomerang sampler as a novel class of continuous-time non-reversible Markov chain Monte Carlo algorithms. The methodology begins by representing the target density as a density, e(−U), with respect to a prescribed (usually) Gaussian measure and constructs a continuous trajectory consisting of a piecewise circular path. The method moves from one circular orbit to another according to a rate function which can be written in terms of U. We demonstrate that the method is easy to implement and demonstrate empirically that it can out-perform existing benchmark piecewise deterministic Markov processes such as the bouncy particle sampler and the Zig-Zag. In the Bayesian statistics context, these competitor algorithms are of substantial interest in the large data context due to the fact that they can adopt data subsampling techniques which are exact (ie induce no error in the stationary distribution). We demonstrate theoretically and empirically that we can also construct a control-variate subsampling boomerang sampler which is also exact, and which possesses remarkable scaling properties in the large data limit. We furthermore illustrate a factorised version on the simulation of diffusion bridges. @en