Hostname: page-component-cb9f654ff-h4f6x Total loading time: 0 Render date: 2025-08-19T00:28:01.149Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Behaviour of the Backward Interpretation of Feynman-Kac Formulae Under Verifiable Conditions

Part of: Parametric inference

Published online by Cambridge University Press:  30 January 2018

Ajay Jasra
Ajay Jasra*
Affiliation:
National University of Singapore
*
∗ Postal address: Department of Statistics and Applied Probability, National University of Singapore, 6 Science Drive 2, 117546, Singapore. Email address: staja@nus.edu.sg
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider the time behaviour associated to the sequential Monte Carlo estimate of the backward interpretation of Feynman-Kac formulae. This is particularly of interest in the context of performing smoothing for hidden Markov models. We prove a central limit theorem under weaker assumptions than adopted in the literature. We then show that the associated asymptotic variance expression for additive functionals grows at most linearly in time under hypotheses that are weaker than those currently existing in the literature. The assumptions are verified for some hidden Markov models.

Keywords

Particle filtercentral limit theoremsmoothing

MSC classification

Secondary: 62F15: Bayesian inference

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 2 , June 2015 , pp. 339 - 359
Copyright
© Applied Probability Trust 

References

Del Moral, P. (2004). Feynman–Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Springer, New York.Google Scholar
Del Moral, P. and Guionnet, A. (2001). On the stability of interacting processes with applications to filtering and genetic algorithms. Ann. Inst. H. Poincaré Prob. Statist., 37, 155–194.Google Scholar
Del Moral, P., Doucet, A. and Singh, S. S. (2010). A backward particle interpretation of Feynman–Kac formulae. M2AN 44, 947–975.Google Scholar
Douc, R. and Moulines, E. (2008). Limit theorems for weighted samples with applications to sequential Monte Carlo methods. Ann. Statist. 36, 2344–2376.Google Scholar
Douc, R., Moulines, E. and Olsson, J. (2014). Long-term stability of sequential Monte Carlo methods under verifiable conditions. Ann. Appl. Prob. 24, 1767–1802.CrossRefGoogle Scholar
Douc, R., Fort, G., Moulines, E. and Priouret, P. (2009). Forgetting the initial distribution for hidden Markov models. Stoch. Process. Appl. 119, 1235–1256.Google Scholar
Douc, R., Garivier, A., Moulines, E. and Olsson, J. (2011). Sequential Monte Carlo smoothing for general state space hidden Markov models. Ann. Appl. Prob. 21, 2109–2145.Google Scholar
Doucet, A., Godsill, S. J. and Andrieu, C. (2000). On sequential Monte Carlo sampling methods for Bayesian filtering. Statist. Comput. 10, 197–208.Google Scholar
Dubarry, C. and Le Corff, S. (2013). Non-asymptotic deviation inequalities for smoothed additive functionals in nonlinear state-space models. Bernoulli 19, 2222–2249.Google Scholar
Godsill, S. J., Doucet, A. and West, M. (2004). Monte Carlo smoothing for nonlinear time series. J. Amer. Statist. Assoc. 99, 156–168.Google Scholar
Poyiadjis, G., Doucet, A. and Singh, S. S. (2011). Particle approximations of the score and observed information matrix in state-space models with application to parameter estimation. Biometrika 98, 65–80.Google Scholar
Whiteley, N. (2013). Stability properties of some particle filters. Ann. Appl. Prob. 23, 2500–2537.Google Scholar