The Application of Zig-Zag Sampler in Sequential Markov Chain Monte Carlo
Particle filtering methods are widely utilized for state estimation in nonlinear non-Gaussian state space models. However, traditional particle filtering methods often suffer from weight degeneracy issues in high-dimensional situations. To overcome this challenge, various particle filtering methods...
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| Published in | Journal of Information Processing Vol. 33; pp. 231 - 244 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Information Processing Society of Japan
2025
一般社団法人 情報処理学会 |
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| Online Access | Get full text |
| ISSN | 1882-6652 1882-6652 |
| DOI | 10.2197/ipsjjip.33.231 |
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| Abstract | Particle filtering methods are widely utilized for state estimation in nonlinear non-Gaussian state space models. However, traditional particle filtering methods often suffer from weight degeneracy issues in high-dimensional situations. To overcome this challenge, various particle filtering methods have been developed to enhance state estimation performance in high-dimensional situations. Among these, the Sequential Markov Chain Monte Carlo (SMCMC) methods, which utilize composite Metropolis-Hastings (MH) kernels, significantly improve state estimation performance. In SMCMC, different Markov Chain Monte Carlo (MCMC) samplers can be incorporated in composite MH kernels as the proposal distribution. Specifically, as a special type of MCMC samplers, the Piecewise Deterministic Markov Process (PDMP) samplers can construct more efficient proposal distributions by utilizing non-reversible Markov processes and have already been applied in the design of composite MH kernels within SMCMC. In this study, we propose a novel approach to further improve state estimation performance in high-dimensional situations. We integrated the Zig-Zag Sampler (ZZS) —a special PDMP sampler—into the SMCMC framework and employed it to construct proposal distributions for the composite MH kernels. This integration fully leverages the advantages of both the ZZS and SMCMC. We assess the efficacy of our proposal method through numerical experiments on challenging high-dimensional state estimation tasks. The results demonstrate that our method significantly improves estimation accuracy and computational efficiency compared to existing state-of-the-art filtering techniques in high-dimensional situations. |
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| AbstractList | Particle filtering methods are widely utilized for state estimation in nonlinear non-Gaussian state space models. However, traditional particle filtering methods often suffer from weight degeneracy issues in high-dimensional situations. To overcome this challenge, various particle filtering methods have been developed to enhance state estimation performance in high-dimensional situations. Among these, the Sequential Markov Chain Monte Carlo (SMCMC) methods, which utilize composite Metropolis-Hastings (MH) kernels, significantly improve state estimation performance. In SMCMC, different Markov Chain Monte Carlo (MCMC) samplers can be incorporated in composite MH kernels as the proposal distribution. Specifically, as a special type of MCMC samplers, the Piecewise Deterministic Markov Process (PDMP) samplers can construct more efficient proposal distributions by utilizing non-reversible Markov processes and have already been applied in the design of composite MH kernels within SMCMC. In this study, we propose a novel approach to further improve state estimation performance in high-dimensional situations. We integrated the Zig-Zag Sampler (ZZS) —a special PDMP sampler—into the SMCMC framework and employed it to construct proposal distributions for the composite MH kernels. This integration fully leverages the advantages of both the ZZS and SMCMC. We assess the efficacy of our proposal method through numerical experiments on challenging high-dimensional state estimation tasks. The results demonstrate that our method significantly improves estimation accuracy and computational efficiency compared to existing state-of-the-art filtering techniques in high-dimensional situations. |
| Author | Han, Yu Nakamura, Kazuyuki |
| Author_xml | – sequence: 1 fullname: Han, Yu organization: Meiji University – sequence: 1 fullname: Nakamura, Kazuyuki organization: Meiji University |
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| Cites_doi | 10.5687/sss.2022.18 10.1007/s10514-009-9130-2 10.1512/iumj.1962.11.11053 10.1007/978-3-319-18347-3_1 10.1109/JSTSP.2015.2497211 10.1007/978-1-4757-3437-9 10.1080/01621459.2017.1294075 10.1109/SSP.2018.8450772 10.1214/18-AOS1715 10.3150/16-BEJ810 10.1109/CAMSAP.2009.5413256 10.1093/biomet/asab013 10.1023/A:1020281327116 10.1016/j.automatica.2012.06.086 10.1109/TSP.2017.2703684 10.1109/TSP.2019.2905816 10.1016/j.spl.2018.02.021 10.1214/18-STS648 10.1201/9780203748039 10.1016/j.jeconom.2015.03.027 10.1016/S0034-4877(16)30031-3 10.1007/s11222-022-10142-x 10.1109/TSIPN.2017.2756563 10.1016/j.dsp.2013.11.006 10.1080/01621459.2022.2099402 10.1109/ICASSP.2017.7952876 10.1007/s11222-015-9598-x 10.1007/978-4-431-55336-6_2 10.1214/18-AAP1453 |
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| References | [18] Hachigian, J. and Rosenblatt, M.: Functions of reversible Markov processes that are Markovian, Journal of Mathematics and Mechanics, pp.951-960 (1962). [21] Li, Y. and Coates, M.: Particle filtering with invertible particle flow, IEEE Transactions on Signal Processing, Vol.65, No.15, pp.4102-4116 (2017). [4] Bierkens, J., Bouchard-Côté, A., Doucet, A., Duncan, A.B., Fearnhead, P., Lienart, T., Roberts, G. and Vollmer, S.J.: Piecewise deterministic Markov processes for scalable Monte Carlo on restricted domains, Statistics & Probability Letters, Vol.136, pp.148-154 (2018). [29] Septier, F., Carmi, A. and Godsill, S.: Tracking of multiple contaminant clouds, 2009 12th International Conference on Information Fusion, pp.1280-1287, IEEE (2009). [30] Septier, F., Pang, S.K., Carmi, A. and Godsill, S.: On MCMC-based particle methods for Bayesian filtering: Application to multitarget tracking, 2009 3rd IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), pp.360-363, IEEE (2009). [3] Bierkens, J.: Non-reversible metropolis-hastings, Statistics and Computing, Vol.26, No.6, pp.1213-1228 (2016). [5] Bierkens, J., Fearnhead, P. and Roberts, G.: The zig-zag process and super-efficient sampling for Bayesian analysis of big data, The Annals of Statistics, Vol.47, No.3, pp.1288-1320 (2019). [6] Bierkens, J., Grazzi, S., Kamatani, K. and Roberts, G.: The boomerang sampler, International Conference on Machine Learning, pp.908-918, PMLR (2020). [14] Davis, M.H.: Markov models & optimization, Routledge (2018). [31] Septier, F. and Peters, G.W.: Langevin and Hamiltonian based sequential MCMC for efficient Bayesian filtering in high-dimensional spaces, IEEE Journal of Selected Topics in Signal Processing, Vol.10, No.2, pp.312-327 (2015). [23] Li, Y., Pal, S. and Coates, M.J.: Invertible particle-flow-based sequential MCMC with extension to Gaussian mixture noise models, IEEE Trans. Signal Processing, Vol.67, No.9, pp.2499-2512 (2019). [10] Carmi, A., Septier, F. and Godsill, S.J.: The Gaussian mixture MCMC particle algorithm for dynamic cluster tracking, Automatica, Vol.48, No.10, pp.2454-2467 (2012). [32] Septier, F. and Peters, G.W.: An overview of recent advances in Monte-Carlo methods for Bayesian filtering in high-dimensional spaces, Theoretical Aspects of Spatial-temporal Modeling, pp.31-61 (2015). [24] Martinez-Cantin, R., De Freitas, N., Brochu, E., Castellanos, J. and Doucet, A.: A Bayesian exploration-exploitation approach for optimal online sensing and planning with a visually guided mobile robot, Autonomous Robots, Vol.27, No.2, pp.93-103 (2009). [1] Andrieu, C., De Freitas, N., Doucet, A. and Jordan, M.I.: An introduction to MCMC for machine learning, Machine Learning, Vol.50, No.1, pp.5-43 (2003). [36] Wu, C. and Robert, C.P.: Generalized bouncy particle sampler, arXiv preprint arXiv:1706.04781 (2017). [25] Mihaylova, L., Carmi, A.Y., Septier, F., Gning, A., Pang, S.K. and Godsill, S.: Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking, Digital Signal Processing, Vol.25, pp.1-16 (2014). [16] Fearnhead, P., Bierkens, J., Pollock, M. and Roberts, G.O.: Piecewise deterministic Markov processes for continuous-time Monte Carlo, Statistical Science, Vol.33, No.3, pp.386-412 (2018). [8] Bierkens, J., Roberts, G.O. and Zitt, P.-A.: Ergodicity of the zigzag process, The Annals of Applied Probability, Vol.29, No.4, pp.2266-2301 (2019). [35] Vanetti, P., Bouchard-Côté, A., Deligiannidis, G. and Doucet, A.: Piecewise-deterministic Markov chain Monte Carlo, arXiv preprint arXiv:1707.05296 (2017). [19] Han, Y. and Nakamura, K.: The Influence of Velocity Refresh in Sequential MCMC with the Invertible Particle Flow and Discrete Bouncy Particle Sampler, Proc. ISCIE International Symposium on Stochastic Systems Theory and its Applications, Vol.2022, pp.18-23, The ISCIE Symposium on Stochastic Systems Theory and Its Applications (2022). [22] Li, Y. and Coates, M.: Sequential MCMC with invertible particle flow, 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp.3844-3848, IEEE (2017). [33] Sherlock, C. and Thiery, A.H.: A discrete bouncy particle sampler, Biometrika, Vol.109, No.2, pp.335-349 (2022). [9] Bouchard-Côté, A., Vollmer, S.J. and Doucet, A.: The bouncy particle sampler: A nonreversible rejection-free Markov chain Monte Carlo method, Journal of the American Statistical Association, Vol.113, No.522, pp.855-867 (2018). [27] Ottobre, M.: Markov chain Monte Carlo and irreversibility, Reports on Mathematical Physics, Vol.77, No.3, pp.267-292 (2016). [34] Van Leeuwen, P.J.: Nonlinear data assimilation for high-dimensional systems, Nonlinear data Assimilation, pp.1-73, Springer (2015). [7] Bierkens, J., Nyquist, P. and Schlottke, M.C.: Large deviations for the empirical measure of the zig-zag process, arXiv preprint arXiv:1912.06635 (2019). [15] Doucet, A., De Freitas, N., Gordon, N.J., et al.: Sequential Monte Carlo methods in practice, Vol.1, No.2, Springer (2001). [17] Goan, E., Perrin, D., Mengersen, K. and Fookes, C.: Piecewise deterministic Markov processes for Bayesian neural networks, Uncertainty in Artificial Intelligence, pp.712-722, PMLR (2023). [20] Lamberti, R., Septier, F., Salman, N. and Mihaylova, L.: Gradient-Based Sequential Markov Chain Monte Carlo for Multitarget Tracking With Correlated Measurements, IEEE Trans. Signal and Information Processing Over Networks, Vol.4, No.3, pp.510-518 (2017). [12] Corbella, A., Spencer, S.E. and Roberts, G.O.: Automatic Zig-Zag sampling in practice, Statistics and Computing, Vol.32, No.6, p.107 (2022). [2] Betancourt, M., Byrne, S., Livingstone, S. and Girolami, M.: The geometric foundations of Hamiltonian Monte Carlo, Bernoulli, Vol.23, No.4A, pp.2257-2298 (2017). [11] Chevallier, A., Fearnhead, P. and Sutton, M.: Reversible jump PDMP samplers for variable selection, Journal of the American Statistical Association, Vol.118, No.544, pp.2915-2927 (2023). [26] Neal, R.M.: Improving asymptotic variance of MCMC estimators: Non-reversible chains are better, arXiv preprint math/0407281 (2004). [13] Creal, D.D. and Tsay, R.S.: High dimensional dynamic stochastic copula models, Journal of Econometrics, Vol.189, No.2, pp.335-345 (2015). [28] Pal, S. and Coates, M.: Sequential MCMC with the discrete bouncy particle sampler, 2018 IEEE Statistical Signal Processing Workshop (SSP), pp.663-667, IEEE (2018). 22 23 24 25 26 27 28 29 30 31 10 32 11 33 12 34 13 35 14 36 15 16 17 18 19 1 2 3 4 5 6 7 8 9 20 21 |
| References_xml | – reference: [19] Han, Y. and Nakamura, K.: The Influence of Velocity Refresh in Sequential MCMC with the Invertible Particle Flow and Discrete Bouncy Particle Sampler, Proc. ISCIE International Symposium on Stochastic Systems Theory and its Applications, Vol.2022, pp.18-23, The ISCIE Symposium on Stochastic Systems Theory and Its Applications (2022). – reference: [9] Bouchard-Côté, A., Vollmer, S.J. and Doucet, A.: The bouncy particle sampler: A nonreversible rejection-free Markov chain Monte Carlo method, Journal of the American Statistical Association, Vol.113, No.522, pp.855-867 (2018). – reference: [10] Carmi, A., Septier, F. and Godsill, S.J.: The Gaussian mixture MCMC particle algorithm for dynamic cluster tracking, Automatica, Vol.48, No.10, pp.2454-2467 (2012). – reference: [21] Li, Y. and Coates, M.: Particle filtering with invertible particle flow, IEEE Transactions on Signal Processing, Vol.65, No.15, pp.4102-4116 (2017). – reference: [18] Hachigian, J. and Rosenblatt, M.: Functions of reversible Markov processes that are Markovian, Journal of Mathematics and Mechanics, pp.951-960 (1962). – reference: [32] Septier, F. and Peters, G.W.: An overview of recent advances in Monte-Carlo methods for Bayesian filtering in high-dimensional spaces, Theoretical Aspects of Spatial-temporal Modeling, pp.31-61 (2015). – reference: [8] Bierkens, J., Roberts, G.O. and Zitt, P.-A.: Ergodicity of the zigzag process, The Annals of Applied Probability, Vol.29, No.4, pp.2266-2301 (2019). – reference: [22] Li, Y. and Coates, M.: Sequential MCMC with invertible particle flow, 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp.3844-3848, IEEE (2017). – reference: [14] Davis, M.H.: Markov models & optimization, Routledge (2018). – reference: [7] Bierkens, J., Nyquist, P. and Schlottke, M.C.: Large deviations for the empirical measure of the zig-zag process, arXiv preprint arXiv:1912.06635 (2019). – reference: [29] Septier, F., Carmi, A. and Godsill, S.: Tracking of multiple contaminant clouds, 2009 12th International Conference on Information Fusion, pp.1280-1287, IEEE (2009). – reference: [4] Bierkens, J., Bouchard-Côté, A., Doucet, A., Duncan, A.B., Fearnhead, P., Lienart, T., Roberts, G. and Vollmer, S.J.: Piecewise deterministic Markov processes for scalable Monte Carlo on restricted domains, Statistics & Probability Letters, Vol.136, pp.148-154 (2018). – reference: [6] Bierkens, J., Grazzi, S., Kamatani, K. and Roberts, G.: The boomerang sampler, International Conference on Machine Learning, pp.908-918, PMLR (2020). – reference: [12] Corbella, A., Spencer, S.E. and Roberts, G.O.: Automatic Zig-Zag sampling in practice, Statistics and Computing, Vol.32, No.6, p.107 (2022). – reference: [20] Lamberti, R., Septier, F., Salman, N. and Mihaylova, L.: Gradient-Based Sequential Markov Chain Monte Carlo for Multitarget Tracking With Correlated Measurements, IEEE Trans. Signal and Information Processing Over Networks, Vol.4, No.3, pp.510-518 (2017). – reference: [31] Septier, F. and Peters, G.W.: Langevin and Hamiltonian based sequential MCMC for efficient Bayesian filtering in high-dimensional spaces, IEEE Journal of Selected Topics in Signal Processing, Vol.10, No.2, pp.312-327 (2015). – reference: [27] Ottobre, M.: Markov chain Monte Carlo and irreversibility, Reports on Mathematical Physics, Vol.77, No.3, pp.267-292 (2016). – reference: [33] Sherlock, C. and Thiery, A.H.: A discrete bouncy particle sampler, Biometrika, Vol.109, No.2, pp.335-349 (2022). – reference: [2] Betancourt, M., Byrne, S., Livingstone, S. and Girolami, M.: The geometric foundations of Hamiltonian Monte Carlo, Bernoulli, Vol.23, No.4A, pp.2257-2298 (2017). – reference: [3] Bierkens, J.: Non-reversible metropolis-hastings, Statistics and Computing, Vol.26, No.6, pp.1213-1228 (2016). – reference: [17] Goan, E., Perrin, D., Mengersen, K. and Fookes, C.: Piecewise deterministic Markov processes for Bayesian neural networks, Uncertainty in Artificial Intelligence, pp.712-722, PMLR (2023). – reference: [5] Bierkens, J., Fearnhead, P. and Roberts, G.: The zig-zag process and super-efficient sampling for Bayesian analysis of big data, The Annals of Statistics, Vol.47, No.3, pp.1288-1320 (2019). – reference: [16] Fearnhead, P., Bierkens, J., Pollock, M. and Roberts, G.O.: Piecewise deterministic Markov processes for continuous-time Monte Carlo, Statistical Science, Vol.33, No.3, pp.386-412 (2018). – reference: [24] Martinez-Cantin, R., De Freitas, N., Brochu, E., Castellanos, J. and Doucet, A.: A Bayesian exploration-exploitation approach for optimal online sensing and planning with a visually guided mobile robot, Autonomous Robots, Vol.27, No.2, pp.93-103 (2009). – reference: [13] Creal, D.D. and Tsay, R.S.: High dimensional dynamic stochastic copula models, Journal of Econometrics, Vol.189, No.2, pp.335-345 (2015). – reference: [30] Septier, F., Pang, S.K., Carmi, A. and Godsill, S.: On MCMC-based particle methods for Bayesian filtering: Application to multitarget tracking, 2009 3rd IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), pp.360-363, IEEE (2009). – reference: [34] Van Leeuwen, P.J.: Nonlinear data assimilation for high-dimensional systems, Nonlinear data Assimilation, pp.1-73, Springer (2015). – reference: [11] Chevallier, A., Fearnhead, P. and Sutton, M.: Reversible jump PDMP samplers for variable selection, Journal of the American Statistical Association, Vol.118, No.544, pp.2915-2927 (2023). – reference: [1] Andrieu, C., De Freitas, N., Doucet, A. and Jordan, M.I.: An introduction to MCMC for machine learning, Machine Learning, Vol.50, No.1, pp.5-43 (2003). – reference: [36] Wu, C. and Robert, C.P.: Generalized bouncy particle sampler, arXiv preprint arXiv:1706.04781 (2017). – reference: [25] Mihaylova, L., Carmi, A.Y., Septier, F., Gning, A., Pang, S.K. and Godsill, S.: Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking, Digital Signal Processing, Vol.25, pp.1-16 (2014). – reference: [35] Vanetti, P., Bouchard-Côté, A., Deligiannidis, G. and Doucet, A.: Piecewise-deterministic Markov chain Monte Carlo, arXiv preprint arXiv:1707.05296 (2017). – reference: [23] Li, Y., Pal, S. and Coates, M.J.: Invertible particle-flow-based sequential MCMC with extension to Gaussian mixture noise models, IEEE Trans. Signal Processing, Vol.67, No.9, pp.2499-2512 (2019). – reference: [28] Pal, S. and Coates, M.: Sequential MCMC with the discrete bouncy particle sampler, 2018 IEEE Statistical Signal Processing Workshop (SSP), pp.663-667, IEEE (2018). – reference: [15] Doucet, A., De Freitas, N., Gordon, N.J., et al.: Sequential Monte Carlo methods in practice, Vol.1, No.2, Springer (2001). – reference: [26] Neal, R.M.: Improving asymptotic variance of MCMC estimators: Non-reversible chains are better, arXiv preprint math/0407281 (2004). – ident: 19 doi: 10.5687/sss.2022.18 – ident: 24 doi: 10.1007/s10514-009-9130-2 – ident: 18 doi: 10.1512/iumj.1962.11.11053 – ident: 34 doi: 10.1007/978-3-319-18347-3_1 – ident: 31 doi: 10.1109/JSTSP.2015.2497211 – ident: 35 – ident: 15 doi: 10.1007/978-1-4757-3437-9 – ident: 9 doi: 10.1080/01621459.2017.1294075 – ident: 28 doi: 10.1109/SSP.2018.8450772 – ident: 5 doi: 10.1214/18-AOS1715 – ident: 2 doi: 10.3150/16-BEJ810 – ident: 30 doi: 10.1109/CAMSAP.2009.5413256 – ident: 7 – ident: 33 doi: 10.1093/biomet/asab013 – ident: 26 – ident: 1 doi: 10.1023/A:1020281327116 – ident: 10 doi: 10.1016/j.automatica.2012.06.086 – ident: 21 doi: 10.1109/TSP.2017.2703684 – ident: 17 – ident: 23 doi: 10.1109/TSP.2019.2905816 – ident: 4 doi: 10.1016/j.spl.2018.02.021 – ident: 16 doi: 10.1214/18-STS648 – ident: 14 doi: 10.1201/9780203748039 – ident: 36 – ident: 13 doi: 10.1016/j.jeconom.2015.03.027 – ident: 27 doi: 10.1016/S0034-4877(16)30031-3 – ident: 29 – ident: 12 doi: 10.1007/s11222-022-10142-x – ident: 20 doi: 10.1109/TSIPN.2017.2756563 – ident: 25 doi: 10.1016/j.dsp.2013.11.006 – ident: 11 doi: 10.1080/01621459.2022.2099402 – ident: 22 doi: 10.1109/ICASSP.2017.7952876 – ident: 6 – ident: 3 doi: 10.1007/s11222-015-9598-x – ident: 32 doi: 10.1007/978-4-431-55336-6_2 – ident: 8 doi: 10.1214/18-AAP1453 |
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| SubjectTerms | high-dimensional filtering non-reversible Markov process piecewise deterministic Markov process Sequential Markov Chain Monte Carlo state space model Zig-Zag Sampler |
| Title | The Application of Zig-Zag Sampler in Sequential Markov Chain Monte Carlo |
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