Exponential ergodicity of the bouncy particle sampler

By George Deligiannidis, Alexandre Bouchard-Cot'e, Arnaud Doucet. Annals of Statistics, 2017

52 citations recorded by the source corpus.

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 \cite{P_dW_12} 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 \cite{J_G_12} 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.

Subfields: Artificial Intelligence, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/05830464bbd81ea4ba017336356eddbcd90feb34

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Citations in the corpus

Geometric ergodicity of the Bouncy Particle Sampler (2018, methodology): “The aim of this paper is to establish geometric convergence to equilibrium for the BPS, in dimension larger than 1, relaxing the conditions given in [10].”
Piecewise deterministic Markov processes and their invariant measures (2018, methodology): “For the BPS process, λ is not in C 1 ( M ) so that we cannot apply the previous theory.”