Exponential ergodicity of the bouncy particle sampler

Abstract

Non-reversible Markov chain Monte Carlo schemes based on piecewise deterministic Markov processes have been recently introduced in applied probability, automatic control, physics and statistics. Although these algorithms demonstrate experimentally good performance and are accordingly increasingly used in a wide range of applications, geometric ergodicity results for such schemes have only been established so far under very restrictive assumptions. We give here verifiable conditions on the target distribution under which the Bouncy Particle Sampler algorithm introduced in [29] is geometrically ergodic. This holds whenever the target satisfies a curvature condition and has tails decaying at least as fast as an exponential and at most as fast as a Gaussian distribution. This allows us to provide a central limit theorem for the associated ergodic averages. When the target has tails thinner than a Gaussian distribution, we propose an original modification of this scheme that is geometrically ergodic. For thick-tailed target distributions, such as t-distributions, we extend the idea pioneered in [19] in a random walk Metropolis context. We apply a change of variable to obtain a transformed target satisfying the tail conditions for geometric ergodicity. By sampling the transformed target using the Bouncy Particle Sampler and mapping back the Markov process to the original parameterization, we obtain a geometrically ergodic algorithm.

The Zig-Zag process and super-efficient sampling for Bayesian analysis of big data

“…Since the first version of this paper was conceived already several other related theoretical and methodological papers have appeared. In particular we mention here results on exponential ergodicity of the BPS (Deligiannidis, Bouchard-Côté and Doucet, 2017) and ergodicity of the multi-dimensional Zig-Zag process (Bierkens, Roberts and Zitt, 2017). The Zig-Zag process has the advantage that it is ergodic under very mild conditions, which in particular means that we are not required to choose a refreshment rate.…”

Strong Invariance Principles for Ergodic Markov Processes

“…for some ε, δ > 0 with stationary measure µ satisfying (27). Then for all f : E → R p with π( f 2+δ ) < ∞, a central limit theorem (CLT) holds:…”

Low-Lying Eigenvalues and Convergence to the Equilibrium of Some Piecewise Deterministic Markov Processes Generators in the Small Temperature Regime

“…Of course there is still a lot of work to do to correctly assess the rate of convergence of such PDMP, * LMBP -Laboratoire de Mathématiques Blaise Pascal, UCA. E-mail: {arnaud.guillin,boris.nectoux}@uca.fr see for example [1,8,12,15], and the behavior with respect to dimension has still to be precisely understood (see however [6]). In practice, the second main advantage of these processes is that they can be used to sample the Gibbs measure (1) without sampling Brownian motions, as for example in Langevin type method such as MALA but only a countable collection of exponentially distributed random variables (using for example thinning procedure).…”

“…On the other hand, if the refreshment is too large compared to λ h,J , the convergence rate towards π ⊗ ν becomes very poor. There is a trade-off between the added refreshment and λ h,J (see for example [12,15] for some very partial explanation). The relevant scaling when h 1 of R v is thus of the same order of λ h,J which scales in h −1 .…”