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

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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


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References

  • [1] Andrieu, C., Doucet, A. and Holenstein, R. (2010). Particle Markov chain Monte Carlo methods. J. R. Stat. Soc. Ser. B. Stat. Methodol. 72 269–342.
  • [2] Andrieu, C., Lee, A. and Vihola, M. Supplement to “Uniform ergodicity of the iterated conditional SMC and geometric ergodicity of particle Gibbs samplers.” DOI:10.3150/15-BEJ785SUPP.
  • [3] Andrieu, C., Lee, A. and Vihola, M. (2013). Uniform ergodicity of the iterated conditional SMC and geometric ergodicity of particle Gibbs samplers. Preprint. Available at arXiv:1312.6432.
  • [4] Andrieu, C. and Roberts, G.O. (2009). The pseudo-marginal approach for efficient Monte Carlo computations. Ann. Statist. 37 697–725.
  • [5] Andrieu, C. and Vihola, M. (2015). Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms. Ann. Appl. Probab. 25 1030–1077.
  • [6] Bérard, J., Del-Moral, P. and Doucet, A. (2013). A lognormal central limit theorem for particle approximations of normalizing constants. Electron. J. Probab. 19.
  • [7] Cérou, F., Del Moral, P. and Guyader, A. (2011). A nonasymptotic theorem for unnormalized Feynman–Kac particle models. Ann. Inst. Henri Poincaré Probab. Stat. 47 629–649.
  • [8] Chopin, N. and Singh, S.S. (2013). On the particle Gibbs sampler. Bernoulli 21 1855–1883.
  • [9] Del Moral, P. (2004). Feynman–Kac Formulae. Probability and Its Applications (New York). New York: Springer.
  • [10] Del Moral, P., Kohn, R. and Patras, F. (2014). On Feynman–Kac and particle Markov chain Monte Carlo models. Preprint. Available at arXiv:1404.5733.
  • [11] Doucet, A., Pitt, M.K., Deligiannidis, G. and Kohn, R. (2015). Efficient implementation of Markov chain Monte Carlo when using an unbiased likelihood estimator. Biometrika 102 295–313.
  • [12] Hobert, J.P. and Robert, C.P. (2004). A mixture representation of $\pi$ with applications in Markov chain Monte Carlo and perfect sampling. Ann. Appl. Probab. 14 1295–1305.
  • [13] Lee, A., Doucet, A. and Łatuszyński, K. (2014). Perfect simulation using atomic regeneration with application to sequential Monte Carlo. Available at arXiv:1407.5770.
  • [14] Lee, A. and Łatuszyński, K. (2014). Variance bounding and geometric ergodicity of Markov chain Monte Carlo kernels for approximate Bayesian computation. Biometrika 101 655–671.
  • [15] Lee, A., Yau, C., Giles, M.B., Doucet, A. and Holmes, C.C. (2010). On the utility of graphics cards to perform massively parallel simulation of advanced Monte Carlo methods. J. Comput. Graph. Statist. 19 769–789.
  • [16] Lindsten, F., Douc, R. and Moulines, E. (2015). Uniform ergodicity of the particle Gibbs sampler. Scand. J. Stat. 42 775–797.
  • [17] Mengersen, K.L. and Tweedie, R.L. (1996). Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24 101–121.
  • [18] Murdoch, D.J. and Green, P.J. (1998). Exact sampling from a continuous state space. Scand. J. Stat. 25 483–502.
  • [19] Propp, J.G. and Wilson, D.B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. In Proceedings of the Seventh International Conference on Random Structures and Algorithms (Atlanta, GA, 1995) 9 223–252. Wiley.
  • [20] Robert, C.P. and Casella, G. (1999). Monte Carlo Statistical Methods. Springer Texts in Statistics. New York: Springer.
  • [21] Roberts, G.O. and Rosenthal, J.S. (1997). Geometric ergodicity and hybrid Markov chains. Electron. Commun. Probab. 2 13–25 (electronic).
  • [22] Roberts, G.O. and Rosenthal, J.S. (2001). Markov chains and de-initializing processes. Scand. J. Stat. 28 489–504.
  • [23] Roberts, G.O. and Tweedie, R.L. (1996). Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika 83 95–110.
  • [24] Sherlock, C., Thiery, A.H., Roberts, G.O. and Rosenthal, J.S. (2015). On the efficiency of pseudo-marginal random walk Metropolis algorithms. Ann. Statist. 43 238–275.
  • [25] Whiteley, N. (2013). Stability properties of some particle filters. Ann. Appl. Probab. 23 2500–2537.

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.