Bayesian Analysis

An Adaptive Sequential Monte Carlo Sampler

Paul Fearnhead and Benjamin M. Taylor

Full-text: Open access

Abstract

Sequential Monte Carlo (SMC) methods are not only a popular tool in the analysis of state–space models, but offer an alternative to Markov chain Monte Carlo (MCMC) in situations where Bayesian inference must proceed via simulation. This paper introduces a new SMC method that uses adaptive MCMC kernels for particle dynamics. The proposed algorithm features an online stochastic optimization procedure to select the best MCMC kernel and simultaneously learn optimal tuning parameters. Theoretical results are presented that justify the approach and give guidance on how it should be implemented. Empirical results, based on analysing data from mixture models, show that the new adaptive SMC algorithm (ASMC) can both choose the best MCMC kernel, and learn an appropriate scaling for it. ASMC with a choice between kernels outperformed the adaptive MCMC algorithm of Haario et al. (1998) in 5 out of the 6 cases considered.

Article information

Source
Bayesian Anal., Volume 8, Number 2 (2013), 411-438.

Dates
First available in Project Euclid: 24 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.ba/1369407558

Digital Object Identifier
doi:10.1214/13-BA814

Mathematical Reviews number (MathSciNet)
MR3066947

Zentralblatt MATH identifier
1208.74095

Keywords
Adaptive MCMC Adaptive Sequential Monte Carlo Bayesian Mixture Analysis Optimal Scaling Stochastic Optimization

Citation

Fearnhead, Paul; Taylor, Benjamin M. An Adaptive Sequential Monte Carlo Sampler. Bayesian Anal. 8 (2013), no. 2, 411--438. doi:10.1214/13-BA814. https://projecteuclid.org/euclid.ba/1369407558


Export citation

References

  • Andrieu, C. and Robert, C. (2001). “Controlled MCMC for Optimal Sampling.” Technical report, Université Paris–Dauphine.
  • Andrieu, C. and Thoms, J. (2008). “A tutorial on adaptive MCMC.” Statistics and Computing, 18(4): 343–373.
  • Atchadé, Y. and Rosenthal, J. (2005). “On adaptive Markov chain Monte Carlo algorithms.” Bernoulli, 11(5): 815–828.
  • Cappé, O., Douc, R., Guillin, A., Marin, J.-M., and Robert, C. P. (2008). “Adaptive importance sampling in general mixture classes.” Statistics and Computing, 18(4): 447–459.
  • Carpenter, J., Clifford, P., and Fearnhead, P. (1999). “Improved particle filter for nonlinear problems.” Radar, Sonar and Navigation, IEEE Proceedings, 146(1): 2 –7.
  • Celeux, G., Hurn, M., and Robert, C. P. (2000). “Computational and Inferential Difficulties with Mixture Posterior Distributions.” Journal of the American Statistical Association, 95(451): 957–970.
  • Chopin, N. (2002). “A sequential particle filter method for static models.” Biometrika, 89(3): 539–552.
  • Cornebise, J., Moulines, E., and Olsson, J. (2008). “Adaptive methods for sequential importance sampling with application to state space models.” Statistics and Computing, 18(4): 461–480.
  • Craiu, R. V., Rosenthal, J., and Yang, C. (2009). “Learn From Thy Neighbor: Parallel-Chain and Regional Adaptive MCMC.” Journal of the American Statistical Association, 104(488): 1454–1466.
  • Crisan, D. (2001). “Particle Filters – A Theoretical perspective.” In Sequential Monte Carlo methods in practice, chapter 2, 17–42. Springer.
  • Del Moral, P. (2004). Feynman-Kac Formulae. Genealogical and interacting particle systems with applications. Springer.
  • Del Moral, P., Doucet, A., and Jasra, A. (2006). “Sequential Monte Carlo samplers.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68(3): 411–436.
  • — (2010). “An Adaptive Sequential Monte Carlo Method for Approximate Bayesian Computation.” URL http://www.cs.ubc.ca/%7Earnaud/delmoral_doucet_jasra_smcabc.pdf
  • Douc, R., Guillin, A., Marin, J.-M., and Robert, C. P. (2007). “Minimum variance importance sampling via Population Monte Carlo.” ESAIM: Probability and Statistics, 11: 427–447.
  • Doucet, A., de Freitas, N., and Gordon, N. (eds.) (2001). Sequential Monte Carlo Methods in Practice. Springer–Verlag New York.
  • Fearnhead, P. (2002). “MCMC, sufficient statistics and particle filters.” Journal of Computational and Graphical Statistics, 11: 848–862.
  • — (2008). “Computational Methods for Complex Stochastic Systems: A Review of Some Alternatives to MCMC.” Statistics and Computing, 18: 151–171.
  • Frühwirth-Schnatter, S. (2006). Finite Mixture and Markov Switching Models. Springer.
  • Gamerman, D. and Lopes, H. F. (2006). Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference (2nd ed.). Chpman & Hall/CRC.
  • Ghosal, S. (1999). “Asymptotic Normality of Posterior Distributions in High Dimensional Linear Models.” Bernoulli, 5(2): 315–331.
  • Gilks, W. and Berzuini, C. (1999). “Following a moving target – Monte Carlo inference for dynamic Bayesian models.” Journal of the Royal Statistical Society, Series B, 63(1): 127–146.
  • Gilks, W., Richardson, S., and Spiegelhalter, D. (eds.) (1995). Markov Chain Monte Carlo in Practice. Chapman & Hall/CRC.
  • Gilks, W. R., Roberts, G. O., and George, E. I. (1994). “Adaptive Direction Sampling.” Journal of the Royal Statistical Society. Series D (The Statistician), 43(1): 179–189.
  • Gordon, N. J., Salmond, D. J., and Smith, A. F. M. (1993). “Novel approach to nonlinear/non-Gaussian Bayesian state estimation.” Radar and Signal Processing, IEEE Proceedings F, 140(2): 107–113.
  • Haario, H., Saksman, E., and Tamminen, J. (1998). “An Adaptive Metropolis algorithm.” Bernoulli, 7: 223–242.
  • Hastings, W. K. (1970). “Monte Carlo sampling methods using Markov chains and their applications.” Biometrika, 57(1): 97–109.
  • Jasra, A., Doucet, A., Stephens, D. A., and Holmes, C. C. (2008a). “Interacting sequential Monte Carlo samplers for trans-dimensional simulation.” Computational Statistics & Data Analysis, 52(4): 1765–1791.
  • Jasra, A., Stephens, D. A., Doucet, A., and Tsagaris, T. (2008b). “Inference for Levy driven Stochastic Volatility Models via Adaptive SMC.” http://www.theodorostsagaris.com/svvg-DAS.pdf.
  • Jasra, A., Stephens, D. A., and Holmes, C. C. (2007). “On population-based simulation for static inference.” Statistics and Computing, 17(3): 263–279.
  • Jennison, C. and Sheehan, N. (1995). “Theoretical and Empirical Properties of the Genetic Algorithm as a Numerical Optimizer.” Journal of Computational and Graphical Statistics, 4(4): 296–318.
  • Kitagawa, G. (1996). “Monte Carlo Filter and Smoother for Non-Gaussian Nonlinear State Space Models.” Journal of Computational and Graphical Statistics, 5(1): 1–25.
  • Kong, A., Liu, J. S., and Wong, W. H. (1994). “Sequential Imputations and Bayesian Missing Data Problems.” Journal of the American Statistical Association, 89(425): 278–288.
  • Liu, J. and West, M. (2001). Sequential Monte Carlo Methods in Practice, chapter 10: Combined Parameter and State Estimation in Simulation-Based Filtering. Springer–Verlag New York.
  • Liu, J. S. and Chen, R. (1995). “Blind Deconvolution Via Sequential Imputations.” Journal of the American Statistical Association, 90: 567–576.
  • — (1998). “Sequential Monte Carlo Methods for Dynamic Systems.” Journal of the American Statistical Association, 93(443): 1032–1044.
  • Liu, J. S., Chen, R., and Wong, W. H. (1998). “Rejection Control and Sequential Importance Sampling.” Journal of the American Statistical Association, 93(443): 1022–1031.
  • Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E. (1953). “Equation of State Calculations by Fast Computing Machines.” The Journal of Chemical Physics, 21(6): 1087–1092.
  • Neal, R. (2001). “Annealed Importance Sampling.” Statistics and Computing, 11(2): 125–139.
  • Pasarica, C. and Gelman, A. (2010). “Adaptively scaling the Metropolis algorithm using expected squared jumped distance.” Statistica Sinica, 20: 343–364.
  • Roberts, G. and Rosenthal, J. (2001). “Optimal Scaling for Various Metropolis-Hastings Algorithms.” Statistical Science, 16(4): 351–367.
  • Roberts, G. O. and Rosenthal, J. S. (2009). “Examples of adaptive MCMC.” Journal of Computational and Graphical Statistics, 18(2): 349–367.
  • Roberts, G. O. and Tweedie, R. L. (1996). “Exponential Convergence of Langevin Distributions and Their Discrete Approximations.” Bernoulli, 2(4): 341–363.
  • Schäfer, C. and Chopin, N. (2013). “Sequential Monte Carlo on large binary sampling spaces.” Statistics and Computing, 23: 163–184.
  • Sherlock, C. and Roberts, G. (2009). “Optimal scaling of the random walk Metropolis on elliptically symmetric unimodal targets.” Bernoulli, 15(3): 774–798.
  • Stephens, M. (2000). “Dealing with label switching in mixture models.” Journal of the Royal Statistical Society, Series B, 62(4): 795–809.
  • Storvik, G. (2002). “Particle filters for state-space models with the presence of unknown static parameters.” IEEE Transactions on Signal Processing, 50: 281–289.
  • Ter Braak, C. J. F. (2006). “A Markov Chain Monte Carlo version of the genetic algorithm Differential Evolution: easy Bayesian computing for real parameter spaces.” Statistics and Computing, 16(3): 239–249.
  • West, M. (1993). “Mixture models, Monte Carlo, Bayesian updating and dynamic models.” Computing Science and Statistics, 24: 325–333.
  • Whitley, D. (1994). “A genetic algorithm tutorial.” Statistics and Computing, 4: 65–85.