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May 2018 Uniform ergodicity of the iterated conditional SMC and geometric ergodicity of particle Gibbs samplers
Christophe Andrieu, Anthony Lee, Matti Vihola
Bernoulli 24(2): 842-872 (May 2018). DOI: 10.3150/15-BEJ785

Abstract

We establish quantitative bounds for rates of convergence and asymptotic variances for iterated conditional sequential Monte Carlo (i-cSMC) Markov chains and associated particle Gibbs samplers [J. R. Stat. Soc. Ser. B. Stat. Methodol. 72 (2010) 269–342]. Our main findings are that the essential boundedness of potential functions associated with the i-cSMC algorithm provide necessary and sufficient conditions for the uniform ergodicity of the i-cSMC Markov chain, as well as quantitative bounds on its (uniformly geometric) rate of convergence. Furthermore, we show that the i-cSMC Markov chain cannot even be geometrically ergodic if this essential boundedness does not hold in many applications of interest. Our sufficiency and quantitative bounds rely on a novel non-asymptotic analysis of the expectation of a standard normalizing constant estimate with respect to a “doubly conditional” SMC algorithm. In addition, our results for i-cSMC imply that the rate of convergence can be improved arbitrarily by increasing $N$, the number of particles in the algorithm, and that in the presence of mixing assumptions, the rate of convergence can be kept constant by increasing $N$ linearly with the time horizon. We translate the sufficiency of the boundedness condition for i-cSMC into sufficient conditions for the particle Gibbs Markov chain to be geometrically ergodic and quantitative bounds on its geometric rate of convergence, which imply convergence of properties of the particle Gibbs Markov chain to those of its corresponding Gibbs sampler. These results complement recently discovered, and related, conditions for the particle marginal Metropolis–Hastings (PMMH) Markov chain.

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Christophe Andrieu. Anthony Lee. Matti Vihola. "Uniform ergodicity of the iterated conditional SMC and geometric ergodicity of particle Gibbs samplers." Bernoulli 24 (2) 842 - 872, May 2018. https://doi.org/10.3150/15-BEJ785

Information

Received: 1 May 2014; Revised: 1 April 2015; Published: May 2018
First available in Project Euclid: 21 September 2017

zbMATH: 06778349
MathSciNet: MR3706778
Digital Object Identifier: 10.3150/15-BEJ785

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

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Vol.24 • No. 2 • May 2018
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