A Discrete Bouncy Particle Sampler

Abstract: Most Markov chain Monte Carlo methods operate in discrete time and are reversible with respect to the target probability. Nevertheless, it is now understood that the use of non-reversible Markov chains can be beneficial in many contexts. In particular, the recently-proposed Bouncy Particle Sampler leverages a continuous-time and non-reversible Markov process and empirically shows state-of-the-art performances when used to explore certain probability densities; however, its implementation typically requires the computation of local upper bounds on the gradient of the log target density. We present the Discrete Bouncy Particle Sampler, a general algorithm based upon a guided random walk, a partial refreshment of direction, and a delayed-rejection step. We show that the Bouncy Particle Sampler can be understood as a scaling limit of a special case of our algorithm. In contrast to the Bouncy Particle Sampler, implementing the Discrete Bouncy Particle Sampler only requires point-wise evaluation of the target density and its gradient. We propose extensions of the basic algorithm for situations when the exact gradient of the target density is not available. In a Gaussian setting, we establish a scaling limit for the radial process as dimension increases to infinity. We leverage this result to obtain the theoretical efficiency of the Discrete Bouncy Particle Sampler as a function of the partial-refreshment parameter, which leads to a simple and robust tuning criterion. A further analysis in a more general setting suggests that this tuning criterion applies more generally. Theoretical and empirical efficiency curves are then compared for different targets and algorithm variations.