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June 2019 Exponential ergodicity of the bouncy particle sampler
George Deligiannidis, Alexandre Bouchard-Côté, Arnaud Doucet
Ann. Statist. 47(3): 1268-1287 (June 2019). DOI: 10.1214/18-AOS1714

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.

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George Deligiannidis. Alexandre Bouchard-Côté. Arnaud Doucet. "Exponential ergodicity of the bouncy particle sampler." Ann. Statist. 47 (3) 1268 - 1287, June 2019. https://doi.org/10.1214/18-AOS1714

Information

Received: 1 August 2017; Revised: 1 April 2018; Published: June 2019
First available in Project Euclid: 13 February 2019

zbMATH: 07053508
MathSciNet: MR3911112
Digital Object Identifier: 10.1214/18-AOS1714

Subjects:
Primary: 60F05 , 60J25 , 65C05 , 65C40

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

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 3 • June 2019
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