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Strong consistency of a modified maximum likelihood estimator for controlled Markov chains

Published online by Cambridge University Press:  14 July 2016

Bharat Doshi  and
Steven E. Shreve
Bharat Doshi*
Affiliation:
Bell Laboratories
Steven E. Shreve*
Affiliation:
Carnegie–Mellon University
*
∗Postal address: HP1B323, Holmdel, NJ 07733, U.S.A. Research carried out when the author was at Rutgers University.
∗∗Postal address: Department of Mathematics, Carnegie-Mellon University, Pittsburgh, PA 15213, U.S.A.
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Abstract

A controlled Markov chain with finite state space has transition probabilities which depend on an unknown parameter α lying in a known finite set A. For each α, a stationary control law ϕ α is given. This paper develops a control scheme whereby at each stage t a parameter α t is chosen at random from among those parameters which nearly maximize the log likelihood function, and the control ut is chosen according to the control law ϕ αt. It is proved that this algorithm leads to identification of the true α under conditions weaker than any previously considered.

Keywords

CONTROLLED MARKOV CHAINIDENTIFICATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 3 , September 1980 , pp. 726 - 734
Copyright
Copyright © Applied Probability Trust 1980 

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Footnotes

Research sponsored in part by the Air Force Office of Scientific Research (AFSC), United States Air Force, under Contract F-49620–79-C-0165.

References

[1] Borkar, V. and Varaiya, P. (1979) Adaptive control of Markov chains, I: Finite parameter set. IEEE Trans. Auto. Control 24, 953–957.Google Scholar
[2] Loève, M. (1960) Probability Theory Van Nostrand, Princeton, NJ.Google Scholar
[3] Mandl, P. (1974) Estimation and control in Markov chains. Adv. Appl. Prob. 6, 40–60.CrossRefGoogle Scholar