Theory of killing and regeneration in continuous-time Monte Carlo sampling

We consider the theory of killing and regeneration for continuous-time Monte Carlo samplers. After a brief introduction in Chapter 1, we begin in Chapter 2 by reviewing some background material relevant to this thesis, including quasi-stationary Monte Carlo methods. These methods are designed to sample from the quasi-stationary distribution of a killed Markov process, and were recently developed to perform scalable Bayesian inference. In Chapter 3 we prove natural sufficient conditions for the quasi-limiting distribution of a killed diffusion to coincide with a target density of interest. We also quantify the rate of convergence to quasi-stationarity by relating the killed diffusion to an appropriate Langevin diffusion. As an example, we consider a killed Ornstein-Uhlenbeck process with Gaussian quasi-stationary distribution. In Chapter 4 we prove convergence of a specific quasi-stationary Monte Carlo method known as `ReScaLE'. We consider the asymptotic behavior of the normalized weighted empirical occupation measures of a diffusion process on a compact manifold which is also killed at a given rate and regenerated at a random location, distributed according to the weighted empirical occupation measure. We show that the weighted occupation measures almost surely comprise an asymptotic pseudo-trajectory for a certain deterministic measure-valued semiflow, after suitably rescaling the time, and that with probability one they converge to the quasi-stationary distribution of the killed diffusion. In Chapter 5 we introduce the Restore sampler. This is a continuous-time sampler, which combines general local dynamics with rebirths from a fixed global rebirth distribution, which occur at a state-dependent rate. In certain settings this rate can be chosen to enforce stationarity of a given target density. The resulting sampler has several desirable properties: simplicity, lack of rejections, regenerations and a potential coupling from the past implementation. The Restore sampler can also be used as a recipe for introducing rejection-free moves into existing MCMC samplers in continuous time. Some simple examples are given to illustrate the potential of Restore. We conclude the thesis in Chapter 6 with some concluding comments and open questions for future work.