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Uniform Ergodicity of the Iterated Conditional SMC and Geometric Ergodicity of Particle Gibbs samplers

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Particle MCMC is an increasingly popular methodology for performing static parameter inference for hidden Markov models on general state spaces, of which one method is the particle Gibbs algorithm. 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. 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. This complements more straightforward results for the particle independent Metropolis—Hastings (PIMH) algorithm. 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. Neither of these phenomena are observed for the PIMH algorithm. 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. These results complement recently discovered, and related, conditions for the particle marginal Metropolis—Hastings (PMMH) Markov chain. This is joint work with Christophe Andrieu and Matti Vihola.

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