The Annals of Statistics

Exponential ergodicity of the bouncy particle sampler

George Deligiannidis, Alexandre Bouchard-Côté, and Arnaud Doucet

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Abstract

Nonreversible 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 [Phys. Rev. E 85 (2012) 026703, 1671–1691] is geometrically ergodic and we provide a central limit theorem for the associated ergodic averages. This holds essentially whenever the target satisfies a curvature condition and the growth of the negative logarithm of the target is at least linear and at most quadratic. For target distributions with thinner tails, we propose an original modification of this scheme that is geometrically ergodic. For targets with thicker tails, we extend the idea pioneered in [Ann. Statist. 40 (2012) 3050–3076] in a random walk Metropolis context. We establish geometric ergodicity of the Bouncy Particle Sampler with respect to an appropriate transformation of the target. Mapping the resulting process back to the original parameterization, we obtain a geometrically ergodic piecewise deterministic Markov process.

Article information

Source
Ann. Statist., Volume 47, Number 3 (2019), 1268-1287.

Dates
Received: August 2017
Revised: April 2018
First available in Project Euclid: 13 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.aos/1550026836

Digital Object Identifier
doi:10.1214/18-AOS1714

Mathematical Reviews number (MathSciNet)
MR3911112

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces 65C05: Monte Carlo methods 65C40: Computational Markov chains 60F05: Central limit and other weak theorems

Keywords
Piecewise deterministic Markov process Markov chain Monte Carlo geometric ergodicity change of variable central limit theorem

Citation

Deligiannidis, George; Bouchard-Côté, Alexandre; Doucet, Arnaud. Exponential ergodicity of the bouncy particle sampler. Ann. Statist. 47 (2019), no. 3, 1268--1287. doi:10.1214/18-AOS1714. https://projecteuclid.org/euclid.aos/1550026836


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Supplemental materials

  • Supplement to “Exponential ergodicity of the bouncy particle sampler”. We provide detailed proofs of all results given in the main paper.