## Bernoulli

- Bernoulli
- Volume 24, Number 2 (2018), 842-872.

### Uniform ergodicity of the iterated conditional SMC and geometric ergodicity of particle Gibbs samplers

Christophe Andrieu, Anthony Lee, and Matti Vihola

#### 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.

#### Article information

**Source**

Bernoulli, Volume 24, Number 2 (2018), 842-872.

**Dates**

Received: May 2014

Revised: April 2015

First available in Project Euclid: 21 September 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1505980880

**Digital Object Identifier**

doi:10.3150/15-BEJ785

**Mathematical Reviews number (MathSciNet)**

MR3706778

**Zentralblatt MATH identifier**

06778349

**Keywords**

geometric ergodicity iterated conditional sequential Monte Carlo Metropolis-within-Gibbs particle Gibbs uniform ergodicity

#### Citation

Andrieu, Christophe; Lee, Anthony; Vihola, Matti. Uniform ergodicity of the iterated conditional SMC and geometric ergodicity of particle Gibbs samplers. Bernoulli 24 (2018), no. 2, 842--872. doi:10.3150/15-BEJ785. https://projecteuclid.org/euclid.bj/1505980880

#### Supplemental materials

- Proofs. The supplementary material contains proofs, intermediate results and a more detailed comparison of the i-cSMC and PGibbs with the PIMH and PMMH.Digital Object Identifier: doi:10.3150/15-BEJ785SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.