Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On particle Gibbs samplers

P. Del Moral, R. Kohn, and F. Patras

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This article analyses a new class of advanced particle Markov chain Monte Carlo algorithms recently introduced by Andrieu, Doucet and Holenstein (J. R. Stat. Soc. Ser. B Stat. Methodol. 72 (2010) 1–269). We present a natural interpretation of these methods in terms of well known unbiasedness properties of Feynman–Kac particle measures, and a new duality with Feynman–Kac models.

This perspective sheds new light on the foundations and the mathematical analysis of this class of methods. A key consequence is their equivalence with the Gibbs sampling of a many-body Feynman–Kac target distribution. Our approach also presents a new stochastic differential calculus based on geometric combinatorial techniques. These techniques allow us to derive non-asymptotic Taylor type series for the semigroup of a class of particle Markov chain Monte Carlo models around their invariant measures with respect to the population size of the auxiliary particle sampler.

These results provide sharp quantitative estimates of the convergence rate to equilibrium of the models with respect to the time horizon and the size of the systems. We illustrate the direct implication of these results with sharp estimates of the contraction coefficient and the Lyapunov exponent of the corresponding samplers, and explicit and non-asymptotic $\mathbb{L}_{p}$-mean error decompositions of the law of the random states around the limiting invariant measure. The abstract framework developed in the article also allows the design of natural extensions to island (also called $\mathrm{SMC}^{2}$) type particle methodologies.

We illustrate this general framework and results in the context of nonlinear filtering, hidden Markov chain problems with fixed unknown parameters, and Feynman–Kac path-integration models arising in computational quantum physics and molecular chemistry.


Cet article analyse une classe de méthodes de Monte Carlo avancées de type particulaire introduites par Andrieu, Doucet, et Holenstein (J. R. Stat. Soc. Ser. B Stat. Methodol. 72 (2010) 1–269). Nous présentons une interprétation naturelle de ces méthodes en termes de mesures de Feynman–Kac particulaires non biaisées classiques et d’une nouvelle formule de dualité entre modèles de Feynman–Kac.

Ce cadre d’étude apporte un nouvel éclairage sur les fondations et l’analyse mathématique de ces méthodes. Une conséquence importante est l’équivalence de ces dernières avec la méthode d’échantillonnage de Gibbs d’une distribution de Feynman–Kac multi-corps. Notre étude développe aussi un nouveau calcul différentiel stochastique fondé sur des techniques géométriques et combinatoires. Ces techniques permettent d’obtenir des développements non asymptotiques des semigroupes de modèles de Monte Carlo par Chaînes de Markov particulaires autour de leur mesure invariante, en fonction de la taille des systèmes de particules en interaction auxiliaires.

Cette analyse conduit à des estimations quantitatives précises de la convergence à l’équilibre de ces modèles par rapport à l’horizon temporel et la taille des systèmes. Nous illustrons ces résultats avec quelques implications directes, notamment l’estimation précise des coefficients de contraction et des exposants de Lyapunov de ces algorithmes de simulation, ainsi que l’estimation fine de l’erreur en norme $\mathbb{L}_{p}$ entre la loi des états aléatoires de ces chaînes de Markov et leur mesure d’équilibre. Le cadre abstrait de l’article permet d’élaborer et d’étendre de façon naturelle ces méthodes à des classes d’algorithmes fondés sur des évolutions d’ilôts particulaires (aussi connus sous le nom $\mathrm{SMC}^{2}$).

Nous montrons enfin comment ce cadre général et les résultats de l’article s’appliquent à l’étude de problèmes de filtrage non linéaire, l’estimation de paramètres fixes dans des modèles de chaînes de Markov cachées, et dans des problèmes d’integration trajectorielle rencontrés en physique quantique et en chimie moléculaire.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 4 (2016), 1687-1733.

Received: 15 October 2014
Revised: 3 June 2015
Accepted: 22 June 2015
First available in Project Euclid: 17 November 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J22: Computational methods in Markov chains [See also 65C40] 65C05: Monte Carlo methods

Particle Gibbs sampling Markov chain Monte Carlo methods Sequential Monte Carlo methods Feynman–Kac models


Del Moral, P.; Kohn, R.; Patras, F. On particle Gibbs samplers. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 4, 1687--1733. doi:10.1214/15-AIHP695.

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  • [1] C. Andrieu, A. Doucet and R. Holenstein. Particle Markov chain Monte Carlo for efficient numerical simulation. In Monte Carlo and Quasi-Monte Carlo Methods 2008 45–60. P. L’Ecuyer and A. B. Owen (Eds). Springer, Berlin, 2009.
  • [2] C. Andrieu, A. Doucet and R. Holenstein. Particle Markov chain Monte Carlo methods (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 72 (3) (2010) 1–269.
  • [3] C. Andrieu, A. Lee and M. Vihola. Uniform ergodicity of the iterated conditional SMC and geometric ergodicity of particle Gibbs samplers. Preprint, 2013. Available at arXiv:1312.6432.
  • [4] R. Arratia and S. DeSalvo. Completely effective error bounds for Stirling numbers of the first and second kinds via Poisson approximation. Preprint, 2014. Available at arXiv:1404.3007.
  • [5] S. Barthelmé and N. Chopin. Expectation propagation for likelihood-free inference. J. Amer. Statist. Assoc. 109 (2014) 315–333.
  • [6] R. Assaraf and M. Caffarel. A pedagogical introduction to quantum Monte Carlo. In Mathematical Models and Methods for Ab Initio Quantum Chemistry 45–73. M. Defranceschi and C. Le Bris (Eds). Lecture Notes in Chemistry 74. Springer, Berlin, 2000.
  • [7] E. Cancès, B. Jourdain and T. Lelièvre. Quantum Monte Carlo simulations of fermions. A mathematical analysis of the fixed-node approximation. Math. Models Methods Appl. Sci. 16 (9) (2006) 1403–1440.
  • [8] O. Cappé, E. Moulines and T. Rydèn. Inference in Hidden Markov Models. Springer, New York, 2005.
  • [9] R. Carmona, P. Del Moral, P. Hu and N. Oudjane. An introduction to particle methods with financial applications. In Numerical Methods in Finance 3–49. Springer Proceedings in Mathematics 12. Springer, New York, 2012.
  • [10] F. Cerou, P. Del Moral and A. Guyader. A non-asymptotic variance theorem for unnormalized Feynman–Kac particle models. Ann. Inst. Henri Poincaré Probab. Stat. 47 (3) (2011) 629–649.
  • [11] H. P. Chan, P. Del Moral and A. Jasra. A sharp first order analysis of Feynman–Kac particle models. Preprint, 2014. Available at arXiv:1411.3800.
  • [12] N. Chopin and S. S. Singh. On particle Gibbs sampling. Bernoulli 21 (2015) 1855–1883.
  • [13] C. Berge. Principles of Combinatorics. Academic Press, New York, 1971.
  • [14] L. Comtet. Analyse Combinatoire. Tomes I, II. Collection SUP. Le Mathématicien 4 5. Presses Universitaires de France, Paris, 1970.
  • [15] D. Creal. A survey of sequential Monte Carlo methods for economics and finance. Econometric Rev. 31 (3) (2012) 245–296.
  • [16] P. Del Moral. Feynman–Kac Formulae. Genealogical and Interacting Particle Systems with Applications. Probability and Its Applications. Springer, New York, 2004.
  • [17] P. Del Moral, M. Ledoux and L. Miclo. On contraction properties of Markov kernels. Probab. Theory Related Fields 126 (2003) 395–420.
  • [18] P. Del Moral and L. Miclo. Branching and interacting particle systems approximations of Feynman–Kac formulae with applications to non-linear filtering. In Séminaire de Probabilités XXXIV 1–145. Lecture Notes in Math. 1729. Springer, Berlin, 2000.
  • [19] P. Del Moral and M. E. Rio. Concentration inequalities for mean field particle models. Ann. Appl. Probab. 21 (3) (2010) 1017–1052.
  • [20] P. Del Moral. Mean Field Simulation for Monte Carlo Integration. Chapman and Hall/CRC Monographs on Statistics and Applied Probability. CRC Press, Boca Raton, FL, 2013.
  • [21] P. Del Moral, F. Patras and S. Rubenthaler. Coalescent tree based functional representations for some Feynman–Kac particle models. Ann. Appl. Probab. 19 (2) (2009) 1–50.
  • [22] P. Del Moral, A. Doucet and S. S. Singh. A backward particle interpretation of Feynman–Kac formulae. M2AN Math. Model. Numer. Anal. 44 (5) (2010) 947–976.
  • [23] P. Del Moral and L. Doucet. Particle motions in absorbing medium with hard and soft obstacles. Stoch. Anal. Appl. 22 (2004) 1175–1207.
  • [24] P. Del Moral and J. Garnier. Genealogical particle analysis of rare events. Ann. Appl. Probab. 15 (4) (2005) 2496–2534.
  • [25] P. Del Moral, P. Hu and L. M. Wu. On the concentration properties of interacting particle processes. Found. Trends Mach. Learn. 3 (3–4) (2012) 225–389.
  • [26] P. Del Moral, P. Jacob, A. Lee, L. Murray and G. W. Peters. Feynman–Kac particle integration with geometric interacting jumps. Stoch. Anal. Appl. 31 (2013) 830–871.
  • [27] P. Del Moral and L. Miclo. On the stability of non linear semigroup of Feynman–Kac type. Ann. Fac. Sci. Toulouse Math. (6) 11 (2002) 135–175.
  • [28] D. N. DeJong, R. Liesenfeld, G. V. Moura, J. F. Richard and H. Dharmarajan. Efficient likelihood evaluation of state-space representations. Rev. Econ. Stud. 80 (2) (2013) 538–567.
  • [29] A. Doucet, J. F. G. de Freitas and N. J. Gordon (Eds). Sequential Monte Carlo Methods in Practice. Springer, New York, 2001.
  • [30] A. Doucet, M. K. Pitt, G. Deligiannidis and R. Kohn. Efficient implementation of Markov chain Monte Carlo when using an unbiased likelihood estimator. Biometrika 102 (2015) 295–313.
  • [31] M. Dowd. Estimating parameters for a stochastic dynamic marine ecological system. Environmetrics 22 (4) (2011) 501–515.
  • [32] M. Dowd and J. Ruth. Estimating behavioral parameters in animal movement models using a state-augmented particle filter. Ecology 92 (3) (2011) 568–575.
  • [33] G. Durham and J. Geweke. Massively parallel Sequential Monte Carlo for Bayesian inference. Unpublished manuscript, 2011. Available at
  • [34] R. G. Everitt. Bayesian parameter estimation for latent Markov random fields and social networks. J. Comput. Graph. Statist. 21 (4) (2012) 940–960.
  • [35] T. Flury and N. Shephard. Bayesian inference based only on simulated likelihood: Particle filter analysis of dynamic economic models. Econometric Theory 27 (5) (2011) 933–956.
  • [36] A. Golightly and D. J. Wilkinson. Bayesian parameter inference for stochastic biochemical network models using particle Markov chain Monte Carlo. Interface Focus 1 (6) (2011) 807–820.
  • [37] S. Henriksen, A. Wills, T. B. Schön and B. Ninness. Parallel implementation of particle MCMC methods on a GPU. In Proceedings of the 16th IFAC Symposium on System Identification 16 1143–1148. Elsevier, Amsterdam, 2012.
  • [38] B. Jourdain, T. Lelièvre and M. El Makrini. Diffusion Monte Carlo method: Numerical analysis in a simple case. M2AN Math. Model. Numer. Anal. 41 (2007) 189–213.
  • [39] N. Kantas, A. Doucet, S. Singh and J. M. Maciejowski. An overview of sequential Monte Carlo methods for parameter estimation in general state space models. In 15th IFAC Symposium on System Identification 115 774–785, 2009.
  • [40] T. Lelièvre, M. Rousset and G. Stoltz. Free Energy Computations: A Mathematical Perspective. Imperial College Press, London, 2010.
  • [41] T. Lelièvre, M. Rousset and G. Stoltz. Computation of free energy differences through non-equilibrium stochastic dynamics: The reaction coordinate case. J. Comput. Phys. 222 (2) (2007) 624–643.
  • [42] F. Lindsten, R. Douc and E. Moulines. Uniform ergodicity of the particle Gibbs sampler. Scand. J. Statist. To appear, 2015.
  • [43] H. Moradkhani, C. M. DeChant and S. Sorooshian. Evolution of ensemble data assimilation for uncertainty quantification using the particle filter Markov chain Monte Carlo method. Water Resour. Res. 48 (2012) W12520.
  • [44] Y. Mishchenko, J. T. T. Vogelstein and L. Paninski. A Bayesian approach for inferring neuronal connectivity from calcium fluorescent imaging data. Ann. Appl. Stat. 5 (2B) (2011) 1229–1261.
  • [45] T. Launay, A. Philippe and S. Lamarche. On particle filters applied to electricity load forecasting. J. SFdS 154 (2013) 1–36.
  • [46] F. Lindsten, T. Schön and M. I. Jordan. Ancestor sampling for particle Gibbs. In Conference in Advances in Neural Information Processing Systems 25 2600–2608. IEEE, New York, 2012.
  • [47] F. Lindsten and T. Schön. On the use of backward simulation in the particle Gibbs sampler. In 37th IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP) 3845–3848. IEEE, New York, 2012.
  • [48] H. F. Lopes and R. S. Tsay. Particle filters and Bayesian inference in financial econometrics. J. Forecast. 30 (1) (2011) 168–209.
  • [49] J. Olsson and T. Ryden. Rao–Blackwellization of particle Markov chain Monte Carlo methods using forward filtering backward sampling. IEEE Trans. Signal Process. 59 (10) (2011) 4606–4619.
  • [50] G. W. Peters, R. G. Hosack and K. R. Hayes. Ecological non-linear state space model selection via adaptive particle Markov chain Monte Carlo (AdPMCMC). Preprint, 2010. Available at arXiv:1005.2238.
  • [51] M. Pitt, R. dos Santos Silva, P. Giordani and R. Kohn. On some properties of Markov chain Monte Carlo simulation methods based on the particle filter. J. Econometrics 171 (2) (2012) 134–151.
  • [52] A. D. Rasmussen, O. Ratmann and K. Koelle. Inference for nonlinear epidemiological models using genealogies and time series. PLoS Comput. Biol. 7 (8) (2011) e1002136.
  • [53] D. Revuz. Markov Chains. North-Holland, Amsterdam, 1975.
  • [54] M. Rousset. On the control of an interacting particle approximation of Schrödinger ground states. SIAM J. Math. Anal. 38 (2006) 824–844.
  • [55] C. Vergé, P. Del Moral, C. Dubarry and E. Moulines. On parallel implementation of sequential Monte Carlo methods: The island particle model. Stat. Comput. 25 (2015) 243–260.
  • [56] J. A. Vrugt, C. J. ter Braak, C. G. Diks and G. Schoups. Hydrologic data assimilation using particle Markov chain Monte Carlo simulation: Theory, concepts and applications. Adv. Water Resour. 51 (2013) 457–478.
  • [57] N. Whiteley, C. Andrieu and A. Doucet. Efficient Bayesian inference for switching state-space models using discrete particle Markov chain Monte Carlo methods. Preprint, 2010. Available at arXiv:1011.2437.