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On the convergence of two sequential Monte Carlo methods for maximum a posteriori sequence estimation and stochastic global optimization

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Abstract

This paper addresses the problem of maximum a posteriori (MAP) sequence estimation in general state-space models. We consider two algorithms based on the sequential Monte Carlo (SMC) methodology (also known as particle filtering). We prove that they produce approximations of the MAP estimator and that they converge almost surely. We also derive a lower bound for the number of particles that are needed to achieve a given approximation accuracy. In the last part of the paper, we investigate the application of particle filtering and MAP estimation to the global optimization of a class of (possibly non-convex and possibly non-differentiable) cost functions. In particular, we show how to convert the cost-minimization problem into one of MAP sequence estimation for a state-space model that is “matched” to the cost of interest. We provide examples that illustrate the application of the methodology as well as numerical results.

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References

  • Ali, M.M., Khompatraporn, C., Zabinsky, Z.B.: A numerical evaluation of several stochastic algorithms on selected continuous global optimization problems. J. Glob. Optim. 31, 635–672 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Anderson, B.D.O., Moore, J.B.: Optimal Filtering. Englewood Cliffs (1979)

  • Bain, A., Crisan, D.: Fundamentals of Stochastic Filtering. Springer, Berlin (2008)

    Google Scholar 

  • Baker, R.D.: How to correctly calculate discounted healthcare costs and benefits. J. Oper. Res. Soc. 51(7), 863–868 (2000)

    MATH  Google Scholar 

  • Bar-Shalom, Y., Blair, W.D. (eds.): Multitarget-Multisensor Tracking: Applications and Advances, vol. III. Artech House, Norwood (2000)

    Google Scholar 

  • Carpenter, J., Clifford, P., Fearnhead, P.: Improved particle filter for nonlinear problems. IEE Proc. Radar Sonar Navig. 146(1), 2–7 (1999)

    Article  Google Scholar 

  • Crisan, D.: Particle filters—a theoretical perspective. In: Doucet, A., de Freitas, N., Gordon, N. (eds.) Sequential Monte Carlo Methods in Practice, pp. 17–42. Springer, Berlin, (2001). Chap 2

    Google Scholar 

  • Crisan, D., Doucet, A.: Convergence of sequential Monte Carlo methods. Technical Report Cambridge University (CUED/FINFENG/TR381) (2000). citeseer.ist.psu.edu/crisan00convergence.html

  • Del Moral, P.: Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Springer, Berlin (2004)

    MATH  Google Scholar 

  • Del Moral, P., Miclo, L.: Branching and interacting particle systems. Approximations of Feynman-Kac formulae with applications to non-linear filtering. In: Lecture Notes in Mathematics, pp. 1–145 (2000)

    Google Scholar 

  • Del Moral, P., Kouritzin, M.A., Miclo, L.: On a class of discrete generation interacting particle systems. Electron. J. Probab. 6(16), 1–26 (2001)

    MathSciNet  Google Scholar 

  • Douc, R., Moulines, E.: Limit theorems for weighted samples with applications to sequential Monte Carlo methods. Ann. Stat. 36(5), 2344–2376 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Douc, R., Cappé, O., Moulines, E.: Comparison of resampling schemes for particle filtering. In: Proceedings of the 4th International Symposium on Image and Signal Processing and Analysis, pp. 64–69 (2005)

    Google Scholar 

  • Doucet, A., Godsill, S., Andrieu, C.: On sequential Monte Carlo Sampling methods for Bayesian filtering. Stat. Comput. 10(3), 197–208 (2000)

    Article  Google Scholar 

  • Doucet, A., de Freitas, N., Gordon, N.: An introduction to sequential Monte Carlo methods. In: Doucet, A., de Freitas, N., Gordon, N. (eds.) Sequential Monte Carlo Methods in Practice, pp. 4–14. Springer, New York (2001a). Chap 1

    Google Scholar 

  • Doucet, A., de Freitas, N., Gordon, N. (eds.): Sequential Monte Carlo Methods in Practice. Springer, New York (2001b)

    MATH  Google Scholar 

  • Du, D., Pardalos, P.M. (eds.): Minimax and Applications. Kluwer Academic, Dordrecht (1995)

    MATH  Google Scholar 

  • Forney, G.D.: The Viterbi algorithm. Proc. IEEE 61(3), 268–278 (1973)

    Article  MathSciNet  Google Scholar 

  • Godsill, S., Doucet, A., West, M.: Maximum a posteriori sequence estimation using Monte Carlo particle filters. Ann. Inst. Stat. Math. 53(1), 82–96 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Godsill, S., Doucet, A., West, M.: Monte Carlo smoothing for nonlinear time series. J. Am. Stat. Assoc. 99(465), 156–168 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Gordon, N., Salmond, D., Smith, A.F.M.: Novel approach to nonlinear and non-Gaussian Bayesian state estimation. IEE Proc., F, Radar Signal Process. 140(2), 107–113 (1993)

    Article  Google Scholar 

  • Gustafsson, F., Gunnarsson, F., Bergman, N., Forssell, U., Jansson, J., Karlsson, R., Nordlund, P.J.: Particle filters for positioning, navigation and tracking. IEEE Trans. Signal Process. 50(2), 425–437 (2002)

    Article  Google Scholar 

  • Heine, K., Crisan, D.: Uniform approximations of discrete-time filters. Adv. Appl. Probab. 40(4), 979–1001 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Hu, X., Schon, T., Ljung, L.: A basic convergence result for particle filtering. IEEE Trans. Signal Process. 56(4), 1337–1348 (2008)

    Article  MathSciNet  Google Scholar 

  • Julier, S.J., Uhlmann, J.: Unscented filtering and nonlinear estimation. Proc. IEEE 92(2), 401–422 (2004)

    Article  Google Scholar 

  • Kalman, R.E.: A new approach to linear filtering and prediction problems. J. Basic Eng. 82, 35–45 (1960)

    Article  Google Scholar 

  • Klaas, M., Lang, D., de Freitas, N.: Fast maximum a posteriori inference in Monte Carlo state spaces. In: Proceedings of the 10th International Workshop on Artificial Intelligence and Statistics (2005)

    Google Scholar 

  • Legland, F.L., Oudjane, N.: Stability and uniform approximation of nonlinear filters using the Hilbert metric and application to particle filters. In: Annals of Applied Probability, pp. 144–187 (2004)

    Google Scholar 

  • Liu, J.S., Chen, R.: Sequential Monte Carlo methods for dynamic systems. J. Am. Stat. Assoc. 93(443), 1032–1044 (1998)

    Article  MATH  Google Scholar 

  • Najim, K., Ikonen, E., Del Moral, P.: Open-loop regulation and tracking control based on a genealogical decision tree. Neural Comput. Appl. 15, 339–349 (2006)

    Article  Google Scholar 

  • Nyblom, P., Olsson, P.M., Rudol, P.: Doherty P Particle filters and map sequence estimation for vehicle tracking (2008). http://www.scientificcommons.org/42403304

  • Pankov, A.R., Platonov, E.N., Semenikhin, K.V.: Minimax optimization of investment portfolio by quantile criterion. Autom. Remote Control 64(7), 1122–1137 (2003). doi:10.1023/A:1024738302885

    Article  MathSciNet  MATH  Google Scholar 

  • Rao, H.I.K., Mathews, V.J., Park, Y.-C.: A minimax approach for the joint design of acoustic cross-talk cancellation filters. IEEE Trans. Audio Speech Lang. Process. 15(8), 2287–2298 (2007)

    Article  Google Scholar 

  • Ristic, B., Arulampalam, S., Gordon, N.: Beyond the Kalman Filter: Particle Filters for Tracking Applications. Artech House, Boston (2004)

    MATH  Google Scholar 

  • Robert, C.P.: The Bayesian Choice. Springer, Berlin (2007)

    MATH  Google Scholar 

  • Saha, S., Boers, Y., Driessen, H., Mandal, P., Bagchi, A.: Particle based MAP state estimation: A comparison. In: Proceedings of the 12th International Conference on Information Fusion. IEEE, pp. 278–283 (2009)

  • Sayed, A.H., Tarighat, A., Khajehnouri, N.: Network based wireless location. IEEE Signal Process. Mag. 22(4), 24–40 (2005)

    Article  Google Scholar 

  • Ziemba, W.T., Vickson, R.G. (eds.): Stochastic Optimization Models in Finance. World Scientific, Singapore (2006)

    MATH  Google Scholar 

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Correspondence to Joaquín Míguez.

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Míguez, J., Crisan, D. & Djurić, P.M. On the convergence of two sequential Monte Carlo methods for maximum a posteriori sequence estimation and stochastic global optimization. Stat Comput 23, 91–107 (2013). https://doi.org/10.1007/s11222-011-9294-4

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  • DOI: https://doi.org/10.1007/s11222-011-9294-4

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