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Simulation from quasi-stationary distributions on reducible state spaces

Part of: Markov processes Nonparametric inference

Published online by Cambridge University Press:  08 September 2017

A. Griffin ,
P. A. Jenkins ,
G. O. Roberts  and
S. E. F. Spencer
A. Griffin*
Affiliation:
University of Warwick
P. A. Jenkins*
Affiliation:
University of Warwick
G. O. Roberts*
Affiliation:
University of Warwick
S. E. F. Spencer*
Affiliation:
University of Warwick
*
* Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK.
* Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK.
* Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK.
* Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK.
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Abstract

Quasi-stationary distributions (QSDs) arise from stochastic processes that exhibit transient equilibrium behaviour on the way to absorption. QSDs are often mathematically intractable and even drawing samples from them is not straightforward. In this paper the framework of sequential Monte Carlo samplers is utilised to simulate QSDs and several novel resampling techniques are proposed to accommodate models with reducible state spaces, with particular focus on preserving particle diversity on discrete spaces. Finally, an approach is considered to estimate eigenvalues associated with QSDs, such as the decay parameter.

Keywords

Quasi-stationary distributionlimiting conditional distributionsimulationresampling methodsequential Monte Carlo

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 62G09: Resampling methods
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 3 , September 2017 , pp. 960 - 980
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Blanchet, J., Glynn, P. and Zheng, S. (2016). Analysis of a stochastic approximation algorithm for computing quasi-stationary distributions. Adv. Appl. Prob. 48, 792–811. Google Scholar
[2] Clancy, D. and Pollett, P. K. (2003). A note on quasi-stationary distributions of birth–death processes and the SIS logistic epidemic. J. Appl. Prob. 40, 821–825. CrossRefGoogle Scholar
[3] Darroch, J. N. and Seneta, E. (1967). On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Prob. 4, 192–196. CrossRefGoogle Scholar
[4] Del Moral, P. (2004). Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Springer, New York. Google Scholar
[5] Del Moral, P., Doucet, A. and Jasra, A. (2006). Sequential Monte Carlo samplers. J. R. Statist. Soc. B 68, 411–436. Google Scholar
[6] 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
[7] Doucet, A., de Freitas, N. and Gordon, N. (eds) (2001). Sequential Monte Carlo Methods in Practice. Springer, New York. Google Scholar
[8] Etheridge, A. (2011). Some Mathematical Models from Population Genetics. Springer, Heidelberg. Google Scholar
[9] Griffin, A. (2016). Quasi-stationary distributions for epidemic models: simulation and characterisation. Doctoral thesis, University of Warwick. Google Scholar
[10] Groisman, P. and Jonckheere, M. (2013). Simulation of quasi-stationary distributions on countable spaces. Markov Process. Relat. Fields 19, 521–542. Google Scholar
[11] Hairer, M., Stuart, A. M. and Vollmer, S. J. (2014). Spectral gaps for a Metropolis–Hastings algorithm in infinite dimensions. Ann. Appl. Prob. 24, 2455–2490. Google Scholar
[12] Jenkins, P. A. (2012). Stopping-time resampling and population genetic inference under coalescent models. Stat. Appl. Genet. Molec. Biol. 11, 9. Google Scholar
[13] Kong, A., Liu, J. S. and Wong, W. H. (1994). Sequential imputations and Bayesian missing data problems. J. Amer. Statist. Assoc. 89, 278–288. Google Scholar
[14] Lambert, A. (2008). Population dynamics and random genealogies. Stoch. Models 24, 45–163. Google Scholar
[15] Lin, M., Chen, R. and Liu, J. S. (2013). Lookahead strategies for sequential Monte Carlo. Statist. Sci. 28, 69–94. CrossRefGoogle Scholar
[16] Liu, J. S. and Chen, R. (1995). Blind deconvolution via sequential imputations. J. Amer. Statist. Assoc. 90, 567–576. Google Scholar
[17] Liu, J. S. and Chen, R. (1998). Sequential Monte Carlo methods for dynamic systems. J. Amer. Statist. Assoc. 93, 1032–1044. Google Scholar
[18] Méléard, S. and Villemonais, D. (2012). Quasi-stationary distributions and population processes. Prob. Surveys 9, 340–410. Google Scholar
[19] Nåsell, I. (1999). On the quasi-stationary distribution of the stochastic logistic epidemic. Math. Biosci. 156, 21–40. Google Scholar
[20] Neal, P. (2014). Endemic behaviour of SIS epidemics with general infectious period distributions. Adv. Appl. Prob. 46, 241–255. Google Scholar
[21] Pollock, M., Fearnhead, P., Johansen, A. M. and Roberts, G. O. (2016). The scalable Langevin exact algorithm: Bayesian inference for big data. Preprint. Available at https://arxiv.org/abs/1609.03436. Google Scholar
[22] Van Doorn, E. A. (2015). Representations for the decay parameter of a birth–death process based on the Courant–Fischer theorem. J. Appl. Prob. 52, 278–289. Google Scholar
[23] Van Doorn, E. A. and Pollett, P. K. (2008). Survival in a quasi-death process. Linear Algebra Appl. 429, 776–791. Google Scholar
[24] Van Doorn, E. A. and Pollett, P. K. (2013). Quasi-stationary distributions for discrete-state models. Europ. J. Operat. Res. 230, 1–14. Google Scholar
[25] Wright, S. (1931). Evolution in Mendelian populations. Genetics 16, 97–159. CrossRefGoogle ScholarPubMed