On the convergence of adaptive sequential Monte Carlo methods. (English) Zbl 1342.82127

The paper studies the consistency and fluctuation properties of a class of adaptive sequential Monte Carlo algorithms. A weak law of large numbers is established such that one can consistently approximate normalizing constants. Central limit theorems are proved to hold at the usual Monte Carlo rate and explicit recursion equations are given for the asymptotic variances. This implies that the fluctuation analysis of the limiting algorithm can be used to describe the asymptotic properties of the adaptive algorithm. The efficiency of the algorithm is illustrated by a numerical application with a complex high-dimensional posterior distribution associated with the Navier-Stokes model.


82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F99 Limit theorems in probability theory
62F15 Bayesian inference
65C05 Monte Carlo methods
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[1] Andrieu, C. and Moulines, É. (2006). On the ergodicity properties of some adaptive MCMC algorithms. Ann. Appl. Probab. 16 1462-1505. · Zbl 1114.65001
[2] Beskos, A., Crisan, D. and Jasra, A. (2014). On the stability of sequential Monte Carlo methods in high dimensions. Ann. Appl. Probab. 24 1396-1445. · Zbl 1304.82070
[3] Beskos, A., Roberts, G. and Stuart, A. (2009). Optimal scalings for local Metropolis-Hastings chains on nonproduct targets in high dimensions. Ann. Appl. Probab. 19 863-898. · Zbl 1172.60328
[4] 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. · Zbl 1233.60047
[5] Cérou, F. and Guyader, A. (2014). Fluctuation analysis of adaptive multilevel splitting. Technical report, INRIA. · Zbl 1362.65018
[6] Chan, H. P. and Lai, T. L. (2013). A general theory of particle filters in hidden Markov models and some applications. Ann. Statist. 41 2877-2904. · Zbl 1293.60071
[7] Chopin, N. (2002). A sequential particle filter method for static models. Biometrika 89 539-551. · Zbl 1036.62062
[8] Chopin, N. (2004). Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. Ann. Statist. 32 2385-2411. · Zbl 1079.65006
[9] Cotter, S. L., Roberts, G. O., Stuart, A. M. and White, D. (2013). MCMC methods for functions: Modifying old algorithms to make them faster. Statist. Sci. 28 424-446. · Zbl 1331.62132
[10] Crisan, D. and Doucet, A. (2000). Convergence of sequential Monte Carlo methods. Technical report, CUED/F-INFENG/, Cambridge Univ.
[11] Del Moral, P. (2004). Feynman-Kac Formulae : Genealogical and Interacting Particle Systems with Applications . Springer, New York. · Zbl 1130.60003
[12] Del Moral, P. (2013). Mean Field Simulation for Monte Carlo Integration . Chapman & Hall, London. · Zbl 1282.65011
[13] Del Moral, P., Doucet, A. and Jasra, A. (2006). Sequential Monte Carlo samplers. J. R. Stat. Soc. Ser. B. Stat. Methodol. 68 411-436. · Zbl 1105.62034
[14] Del Moral, P., Doucet, A. and Jasra, A. (2012). On adaptive resampling strategies for sequential Monte Carlo methods. Bernoulli 18 252-278. · Zbl 1236.60072
[15] Del Moral, P., Doucet, A. and Jasra, A. (2012). An adaptive sequential Monte Carlo method for approximate Bayesian computation. Stat. Comput. 22 1009-1020. · Zbl 1252.65025
[16] Douc, R. and Moulines, E. (2008). Limit theorems for weighted samples with applications to sequential Monte Carlo methods. Ann. Statist. 36 2344-2376. · Zbl 1155.62056
[17] Douc, R., Moulines, E. and Olsson, J. (2014). Long-term stability of sequential Monte Carlo methods under verifiable conditions. Ann. Appl. Probab. 24 1767-1802. · Zbl 1429.62364
[18] Doucet, A. and Johansen, A. (2011). A tutorial on particle filtering and smoothing: Fifteen years later. In Handbook of Nonlinear Filtering (D. Crisan and B. Rozovsky, eds.). Oxford Univ. Press, Oxford. · Zbl 05919872
[19] Gelman, A. and Meng, X.-L. (1998). Simulating normalizing constants: From importance sampling to bridge sampling to path sampling. Statist. Sci. 13 163-185. · Zbl 0966.65004
[20] Giraud, F. and Del Moral, P. (2015). Non-asymptotic analysis of adaptive and annealed Feynman-Kac particle models. Bernoulli .
[21] Jasra, A., Stephens, D. A., Doucet, A. and Tsagaris, T. (2011). Inference for Lévy-driven stochastic volatility models via adaptive sequential Monte Carlo. Scand. J. Stat. 38 1-22. · Zbl 1246.91149
[22] Kantas, N., Beskos, A. and Jasra, A. (2014). Sequential Monte Carlo methods for high-dimensional inverse problems: A case study for the Navier-Stokes equations. SIAM/ASA J. Uncertain. Quantificat. 2 464-489. · Zbl 1308.65010
[23] Schäfer, C. and Chopin, N. (2013). Sequential Monte Carlo on large binary sampling spaces. Stat. Comput. 23 163-184. · Zbl 1322.62035
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