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Uniform ergodicity of the iterated conditional SMC and geometric ergodicity of particle Gibbs samplers
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Andrieu, Christophe , Lee, Anthony and Vihola, Matti (2017) Uniform ergodicity of the iterated conditional SMC and geometric ergodicity of particle Gibbs samplers. Bernoulli, 24 (2). pp. 842-872. doi:10.3150/15-BEJ785 ISSN 1350-7265.
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Official URL: http://doi.org/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 NN, 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 NN 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.
| Item Type: | Journal Article | |||||||||||||||
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| Subjects: | Q Science > QA Mathematics | |||||||||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | |||||||||||||||
| Library of Congress Subject Headings (LCSH): | Monte Carlo method, Bayesian statistical decision theory, Markov processes, Mathematical statistics, Stochastic geometry | |||||||||||||||
| Journal or Publication Title: | Bernoulli | |||||||||||||||
| Publisher: | Int Statistical Institute | |||||||||||||||
| ISSN: | 1350-7265 | |||||||||||||||
| Official Date: | 21 September 2017 | |||||||||||||||
| Dates: |
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| Volume: | 24 | |||||||||||||||
| Number: | 2 | |||||||||||||||
| Page Range: | pp. 842-872 | |||||||||||||||
| DOI: | 10.3150/15-BEJ785 | |||||||||||||||
| Status: | Peer Reviewed | |||||||||||||||
| Publication Status: | Published | |||||||||||||||
| Access rights to Published version: | Restricted or Subscription Access | |||||||||||||||
| Date of first compliant deposit: | 21 March 2017 | |||||||||||||||
| Date of first compliant Open Access: | 27 September 2017 | |||||||||||||||
| RIOXX Funder/Project Grant: |
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