Abstract
Consideration was given to the minimax estimation in the observation system including a hidden Markov model for continuous and counting observations. The dynamic and observation equations depend on a random finite-dimensional parameter having an unknown distribution with the given support. The conditional expectation of the available observation of some generalized quadratic loss function was used as the risk function. Existence of the saddle point in the formulated minimax problem was proved, and the worst distribution and the minimax estimate as the solution of a simpler dual problem were characterized.
Similar content being viewed by others
References
Kats, I.Ya. and Kurzhanskii, A.B., Minimax Multistep Filtration in the Statistically Uncertain Situations, Avtom. Telemekh., 1978, no. 11, pp. 79–87.
Martin, C.J. and Mintz, M., Robust Filtering and Prediction for Linear Systems with Uncertain Dynamics: A Game-Theoretic Approach, IEEE Trans. Autom. Control, 1983, vol. 9, pp. 888–896.
Anan’ev, B.I., Minimax Linear Filtration of the Multistep Processs with Uncertain Distribution of Perturbations, Avtom. Telemekh., 1993, no. 10, pp. 131–139.
Elliott, R.J., Aggoun, L., and Moore, J.B., Hidden Markov Models: Estimation and Control, Berlin: Springer, 1995.
Borisov, A.V., Preliminary Analysis of the Distribution of States of the Randomly Structured Special-purpose Controlled Systems, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2005, no. 1, pp. 48–62.
Liptser, R.Sh. and Shiryaev, A.N., Teoriya martingalov, Moscow: Nauka, 1986. Translated under the title Theory of Martingales, Dordrecht: Kluwer, 1989.
Loève, M., Probability Theory, Princeton, New Jersey: Van Nostrand, 1960. Translated under the title Teoriya veroyatnostei, Moscow: Mir, 1962.
Wong, E. and Hajek, B., Stochastic Processes in Engineering Systems, New York: Springer, 1985.
Liptser, R.Sh. and Shiryaev, A.N., Statistika sluchainykh protsessov, Moscow: Nauka, 1974. Translated into English under the title Statistics of Random Processes, Berlin: Springer, 1978.
Borisov, A.V., Analysis of the States of the Hidden Markov Processes Generated by the Special Jump Processes, Teor. Veroyatn. Primen., 2006, no. 3, pp. 589–600.
Ladyzhenskaya, O.A., Solonnikov, V.A., and Ural’tseva, N.N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa (Linear and Quasilinear Parabolic Equations), Moscow: Nauka, 1967.
Pardoux, E., Stochastic Partial Differential Equations and Filtering of Diffusion Processes, Stochastics, 1979, vol. 3, pp. 127–167.
Pardoux, E., Filtering of a Diffusion Process with Poisson Type Observation, in Stochastic Control Theory and Stochastic Differential Equations. Lecture Notes in Control and Information Sciences, 1979, vol. 16.
Semenikhin, K.V., Minimax Estimation of the Random Elements by the RMS Criterion, Teor. Veroyatn. Primen., 2003, no. 5, pp. 12–25.
Author information
Authors and Affiliations
Additional information
Original Russian Text © A. V. Borisov, 2007, published in Avtomatika i Telemekhanika, 2007, No. 11, pp. 31–45.
This work was supported by the Russian Foundation for Basic Research, project no. 05-01-00508-a, and the Program for Fundamental Algorithms of Information Technologies of the Department of Information Technologies and Computer Systems, Russian Academy of Sciences.
Rights and permissions
About this article
Cite this article
Borisov, A.V. Minimax a posteriori estimation in the hidden Markov models. Autom Remote Control 68, 1917–1930 (2007). https://doi.org/10.1134/S0005117907110033
Received:
Issue Date:
DOI: https://doi.org/10.1134/S0005117907110033