Robust Recursive Bayesian Estimation and Quantum Minimax Strategies

  • P. Pardalos
  • V. Yatsenko
  • S. Butenko
Part of the Applied Optimization book series (APOP, volume 66)


The problem of a recursive realization of Bayesian estimation for incomplete experimental data is considered. A differential-geometric structure of nonlinear estimation is studied. It is shown that the use of a rationally chosen description of the true posterior density produces a geometrical structure defined on the family of possible posteriors. Pythagorean-like relations valid for probability distributions are presented and their importance for estimation under reduced data is indicated. A robust algorithm for estimation of unknown parameters is proposed, which is based on a quantum implementation of the Bayesian estimation procedure.

Key words

Bayesian estimation robust estimation control optimization quantum detection model approximation 


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  1. [1]
    Amari, S.-I., Differential geometry of curved exponential families-curvatures and information loss. The Annals of Statistics, 10(2), pp. 357–385, 1982.MathSciNetzbMATHGoogle Scholar
  2. [2]
    Amari, S.-I., Differential-Geometrical Methods in Statistics, Springer-Verlag, New York, 1985.Google Scholar
  3. [3]
    Belavkin V. P., O. Hirota, and R.L. Hudson, Quantum communication and measurement, Plenum Press, 1996.Google Scholar
  4. [4]
    C. Bendjaballah, O. Hirota, and S. Reynaud, Quantum Aspect of Optical Communications, Lecture Note in Physics, Springer, LNP-378, 1991.Google Scholar
  5. [5]
    Boothby, W. M., An Introduction to Differential Manifolds and Riemannian Geometry, Academic Press, London, 1975.Google Scholar
  6. [6]
    Chentsov, N. N., Statistical Decision Rules and Optimal Inference (in Russian), Nauka, Moscow, 1972. English translation: Translation of Mathematical Monographs, 53. AMS, Rhode Island, 1982.Google Scholar
  7. [7]
    Golodnikov, A., P. Knopov, P. Pardalos and S. Uryasev, “Optimization in the space of distribution functions and applications in the Bayes Analysis”. In “Probabilistic Constrained Optimization” (S. Uryasev, Editor), pp. 102–131, Kluwer Academic Publishers, 2000.Google Scholar
  8. [8]
    Helstrom C.W., Quantum Detection and Estimation Theory, Academic Press, 1976.Google Scholar
  9. [9]
    Insua, D. R. and F. Ruggeri, Eds., Robust Bayesian Analysis, Lecture Notes in Statistics, Springer Verlag, 2000.Google Scholar
  10. [10]
    Kobayashi, S. and K. Nomixu, Foundations of Differential Geometry, Vol. I, Interscience Publishers, New York, 1963.Google Scholar
  11. [11]
    Kulhavý, R., Recursive Nonlinear Estimation: a Geometric Approach, Springer, London, 1996.Google Scholar
  12. [12]
    Kulhavý, R., Differential geometry of recursive nonlinear estimation. In Preprints of the 11th IFAC World Congress, Tallinn, Estonia. Vol. 3, pp. 113–118, 1990.Google Scholar
  13. [13]
    Peterka, V., Bayesian approach to system identification. In P. Eyi-choff (ed.), Trends and Progress in System Identification, Pergamon Press, Oxford, 1981.Google Scholar
  14. [14]
    Savage, L. J., The Foundations of Statistics, Wiley, New York, 1954.Google Scholar
  15. [15]
    Sorenson, H. W., Recursive estimation for nonlinear dynamic systems, in J. C. Spall (Ed.), Bayesian Analysis of Time Series and Dynamic Models, Marcel Dekker, New York, 1988.Google Scholar
  16. [16]
    Yatsenko, V., T. Titarenko, Yu. Kolesnik, Identification of the non-Gaussian chaotic dynamics of the radioemission back scattering processes, Proc. Of the 10th IFAC Simposium on System Identification SYSID’94 Kobenhaven, 4–6 July, 1994.-V. 1.-P. 313–317.Google Scholar
  17. [17]
    Wets, R., Statistical estimation from an optimization viewpoint, Annals of Operations Research, 85, pp.79–101, 1999.CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • P. Pardalos
    • 1
  • V. Yatsenko
    • 2
  • S. Butenko
    • 3
  1. 1.Center for Applied Optimization, Dept. of Industrial and Systems EngineeringUniversity of FloridaUSA
  2. 2.Scientific Foundation of Researchers and Specialists on Molecular Cybernetics and InformaticsUSA
  3. 3.Dept. of Industrial and Systems EngineeringUniversity of FloridaUSA

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