Piecewise-deterministic Markov chain Monte Carlo

Thesis

Recent interest in a class of Markov chain Monte Carlo schemes based on continuous-time piecewise-deterministic Markov processes has led to several new and promising algorithmic developments. Prominent examples include the zig-zag process and the bouncy particle sampler. We explore this class of algorithms, proposing extensions and drawing connections to existing literature.

Two key aspects of a continuous-time piecewise-deterministic process are the flow and the event kernel. We discuss conditions on these sufficient to design a process targeting a distribution of interest. Based on these conditions, we show how the flow can be extended to Hamiltonian dynamics when exactly computable. For the event kernel, we demonstrate that a simple factorization of the invariant distribution subsumes all existing kernels and allows new kernels to be easily derived.

Next, we explore the idea of piecewise-deterministic processes in discrete time. These algorithms have many close connections with pre-existing algorithms. The simultaneous simulation of multiple event rates may be connected to the discrete-time delayed acceptance algorithm. We demonstrate that the flow and event kernels used in the bouncy particle sampler bear strong resemblance to the reflective slice sampler. A class of discrete-time event kernels which we call here tunnelling algorithms also prove to be a special case of this framework.

Applying this framework to the setting of large datasets with many observations, we describe how both the continuous-time and discrete-time algorithms can be adapted to this setting. We explore the use of approximations to bound the effects of individual observations. Methods of simulating from discrete distributions based on Poisson processes are used to efficiently subsample the data. Finally, we show how these methods can be used to extend and improve upon the existing firefly Monte Carlo algorithm.

A second application is found in the setting where interaction between parameters is limited and forms a sparse graph, for which the continuous-time local bouncy particle sampler algorithm has been developed. A discrete-time extension mimicking the local properties of this algorithm is proposed, unveiling a close connection to the random cluster algorithm of Swendsen-Wang.

Authors: Vanetti, P

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: big data; simulation; Markov chain Monte Carlo; random cluster methods; piecewise-deterministic Markov processes; Monte Carlo; Bayesian statistics; Swendsen-Wang; logistic regression
• Subjects: Bayesian statistics