Abstract
We develop in this paper three multiple-try blocking schemes for Bayesian analysis of nonlinear and non-Gaussian state space models. To reduce the correlations between successive iterates and to avoid getting trapped in a local maximum, we construct Markov chains by drawing state variables in blocks with multiple trial points. The first and second methods adopt autoregressive and independent kernels to produce the trial points, while the third method uses samples along suitable directions. Using the time series structure of the state space models, the three sampling schemes can be implemented efficiently. In our multimodal examples, the three multiple-try samplers are able to generate the desired posterior sample, whereas existing methods fail to do so.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrade Netto M. L., Gimeno L., and Mendes M. J. 1978. On the optimal and suboptimal nonlinear filtering problem for discrete-time systems. IEEE Transactions on Automatic Control 23: 1062–67.
Besag J., Green P., Higon D. and Mengersen K. 1995. Bayesian computation and stochastic systems (with discussion). Statistical Science, 10: 3–66.
Box G. E. P. and Jenkins G. M. 1976. Time Series Analysis: Forecasting and Control. San Francisco, CA: Holden-Day.
Carlin B. P., Polson N. G. and Stoffer D. 1992. A Monte Carlo approach to nonnormal and nonlinear state-space modeling. Journal of the American Statistical Association 87: 493–500.
Carter C. K. and Kohn R. 1994. On Gibbs sampling for state space models. Biometrika 81: 541–53.
De Jong P. and Shephard N. 1995. The simulation smoother for time series models. Biometrika 82: 339–50.
Durbin J. and Koopman S. J. 2000. Time series analysis of non-Gaussian observations based on state space models from both classical and Bayesian perspectives. Journal of the Royal Statistical Society B 62: 3–29.
Gelman A., Roberts G. O. and Gilks W. R. 1996. Efficient Metropolis jumping rules. In Bayesian Statistics 5, Ed. J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, pp. 599–607. New York: Oxford University Press.
Gerlach R., Carter C. and Kohn R. 2000. Efficient Bayesian inference for dynamic mixture models. Journal of the American Statistical Association 95: 819–828.
Geweke J. and Tanizaki H. 2000. On Markov chain Monte Carlo methods for nonlinear and non-Gaussian state-space models. Communications in Statistics–-Simulation and Computation 28: 867–94.
Gilks W. R., Roberts G. O. and George E. I. 1994. Adaptive direction sampling.The Statistician 43: 179–89.
S. J. Godsill A. Doucet M. West (2004) ArticleTitleMonte Carlo smoothing for nonlinear time series Journal of the American Statistical Association 99 156–168 Occurrence Handle2054295
Green P. J. and Han X. L. 1990. Metropolis methods, Gaussian proposals and antithetic variables. In Stochastic Models, Statistical Methods and Algorithms in Image Analysis, Ed. P. Barone, A. Frigessi and M. Piccioni, pp. 142–164. Berlin: Springer-Verlag.
A. C. Harvey, 1989 Forecasting Structural Time Series Models and the Kalman Filter. Cambridge: Cambridge University Press.
Jacquier E., Polson N. G. and Rossi P. E. 1994. Bayesian analysis of stochastic volatility models (with discussion). Journal of Business & Economic Statistics 12: 371–417.
Kitagawa G. 1987. Non-Gaussian state-space modeling of nonstationary time series.Journal of the American Statistical Association 82: 1032–63.
Kitagawa G. 1996. Monte Carlo filter and smoother for non-Gaussian nonlinear state-space models. Journal of Computational and Graphical Statistics 5: 1–25.
Liu J., Liang F. and Wong W. H. 2000. The multiple-try method and local optimization in Metropolis sampling. Journal of the American Statistical Association 95: 121–34.
Liu J., Wong W. H. and Kong A. 1994. Covariance structure of the Gibbs sampler with applications to the comparisons of estimators and augmentation schemes. Biometrika 81: 27–40.
Roberts G. O. and Gilks W. R. 1994. Convergence of adaptive direction sampling. Journal of Multivariate Analysis 49: 287–98.
Shephard N. 1994. Partial non-Gaussian state space. Biometrika 81: 115–31.
Shephard N. 1996. Statistical aspect of ARCH and stochastic volatility. In Time Series Models in Econometrics, Finance and other Fields, Ed. D. R. Cox, D. V. Hinkley and O. E. Barndorff-Nielsen, pp. 1–67. London: Chapman and Hall.
Shephard N. and Pitt M. K. 1997. Likelihood analysis of non-Gaussian measurement time series. Biometrika 84: 653–67.
So M. K. P. 1999. Time series with additive noise. Biometrika 86: 474–482.
So M. K. P. 2003. Posterior mode estimation for nonlinear and non-Gaussian state space models. Statistica Sinica 13: 255–274.
Stroud J. R., Muller P., and Polson N. G. 2003. Nonlinear state-space models with state-dependent variances. Journal of the American Statistical Association 98: 377–386.
Tanizaki H. 2004. Computational Methods in Statistics and Econometrics, (Statistics: Textbooks and Monographs, Vol. 172), Mercel Dekker.
Tierney L. 1994. Markov chains for exploring posterior distributions. Annals of Statistics 22: 1701–1762
West M. and Harrison P. J. 1997. Bayesian Forecasting and Dynamic Models. New York: Springer-Verlag.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
So, M.K.P. Bayesian analysis of nonlinear and non-Gaussian state space models via multiple-try sampling methods. Stat Comput 16, 125–141 (2006). https://doi.org/10.1007/s11222-006-6891-8
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11222-006-6891-8