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Sensitivity of hidden Markov models

Part of: Inference from stochastic processes Controllability, observability, and system structure Markov processes Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Alexander Yu. Mitrophanov ,
Alexandre Lomsadze  and
Mark Borodovsky
Alexander Yu. Mitrophanov*
Affiliation:
Georgia Institute of Technology
Alexandre Lomsadze*
Affiliation:
Georgia Institute of Technology
Mark Borodovsky*
Affiliation:
Georgia Institute of Technology
*
∗Postal address: School of Biology, Georgia Institute of Technology, 310 Ferst Drive, Atlanta, GA 30332-0230, USA.
∗Postal address: School of Biology, Georgia Institute of Technology, 310 Ferst Drive, Atlanta, GA 30332-0230, USA.
∗∗Postal address: School of Biomedical Engineering, Georgia Institute of Technology, 313 Ferst Drive, Atlanta, GA 30332-0535, USA. Email address: mark@amber.biology.gatech.edu
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Abstract

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We derive a tight perturbation bound for hidden Markov models. Using this bound, we show that, in many cases, the distribution of a hidden Markov model is considerably more sensitive to perturbations in the emission probabilities than to perturbations in the transition probability matrix and the initial distribution of the underlying Markov chain. Our approach can also be used to assess the sensitivity of other stochastic models, such as mixture processes and semi-Markov processes.

Keywords

Hidden Markov modelMarkov chainmixturesemi-Markov processperturbation boundsensitivity analysis

MSC classification

Primary: 93B35: Sensitivity (robustness)
Secondary: 62M09: Non-Markovian processes 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60E05: Distributions
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 3 , September 2005 , pp. 632 - 642
Copyright
© Applied Probability Trust 2005 

References

Archer, G. E. B. and Titterington, D. M. (2002). Parameter estimation for hidden Markov chains. J. Statist. Planning Infer. 108, 365–390.CrossRefGoogle Scholar
Ball, F., Milne, R. K. and Yeo, G. F. (1991). Aggregated semi-Markov processes incorporating time interval omission. Adv. Appl. Prob. 23, 772–797.CrossRefGoogle Scholar
Baum, L. E. and Petrie, T. (1966). Statistical inference for probabilistic functions of finite Markov chains. Ann. Math. Statist. 37, 1554–1563.CrossRefGoogle Scholar
Baum, L. E., Petrie, T., Soules, G. and Weiss, N. (1970). A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. Ann. Math. Statist. 41, 164–171.CrossRefGoogle Scholar
Besemer, J. and Borodovsky, M. (1999). Heuristic approach to deriving models for gene finding. Nucleic Acids Res. 27, 3911–3920.CrossRefGoogle ScholarPubMed
Besemer, J., Lomsadze, A. and Borodovsky, M. (2001). GeneMarkS: a self-training method for prediction of gene starts in microbial genomes. Implications for finding sequence motifs in regulatory regions. Nucleic Acids Res. 29, 2607–2618.CrossRefGoogle ScholarPubMed
Bickel, P. G., Ritov, Y. and Rydén, T. (1998). Asymptotic normality of the maximum-likelihood estimator for general hidden Markov models. Ann. Statist. 26, 1614–1635.CrossRefGoogle Scholar
Cho, G. E. and Meyer, C. D. (2001). Comparison of perturbation bounds for the stationary distribution of a Markov chain. Linear Algebra Appl. 335, 137–150.CrossRefGoogle Scholar
Cohen, J. E., Kemperman, J. H. B. and Zbăganu, Gh. (1998). Comparisons of Stochastic Matrices. Birkhäuser, Boston, MA.Google Scholar
Ephraim, Y. and Merhav, N. (2002). Hidden Markov processes. IEEE Trans. Inf. Theory 48, 1518–1569.CrossRefGoogle Scholar
Granovsky, B. L. and Zeifman, A. I. (2000). Nonstationary Markovian queues. J. Math. Sci. (New York) 99, 1415–1438.CrossRefGoogle Scholar
Jensen, J. L. and Petersen, N. V. (1999). Asymptotic normality of the maximum likelihood estimator in state space models. Ann. Statist. 27, 514–535.CrossRefGoogle Scholar
Juang, B.-H. and Rabiner, L. R. (1985). A probabilistic distance measure for hidden Markov models. AT&T Tech. J. 64, 391–408.Google Scholar
Kartashov, N. V. (1996). Strong Stable Markov Chains. VSP, Utrecht.CrossRefGoogle Scholar
Koski, T. (2001). Hidden Markov Models for Bioinformatics. Kluwer, Dordrecht.CrossRefGoogle Scholar
Leroux, B. G. (1992). Maximum likelihood estimation for hidden Markov models. Stoch. Process. Appl. 40, 127–143.CrossRefGoogle Scholar
Lukashin, A. L. and Borodovsky, M. (1998). GeneMark.hmm: new solutions for gene finding. Nucleic Acids Res. 26, 1107–1115.CrossRefGoogle ScholarPubMed
McDonald, I. and Zucchini, W. (1997). Hidden Markov and Other Models for Discrete-Valued Time Series. Chapman and Hall, London.Google Scholar
Mitrophanov, A. Yu. (2003). Stability and exponential convergence of continuous-time Markov chains. J. Appl. Prob. 40, 970–979.CrossRefGoogle Scholar
Mitrophanov, A. Yu. (2004). The spectral gap and perturbation bounds for reversible continuous-time Markov chains. J. Appl. Prob. 41, 1219–1222.CrossRefGoogle Scholar
Mitrophanov, A. Yu. (2005). Ergodicity coefficient and perturbation bounds for continuous-time Markov chains. Math. Ineq. Appl. 8, 159–168.Google Scholar
Mitrophanov, A. Yu. (2005). Estimates of sensitivity to perturbations for finite homogeneous continuous-time Markov chains. Teor. Veroyat. Primen. 50, 371–379 (in Russian). English translation to appear in Theory Prob. Appl. Google Scholar
Mitrophanov, A. Yu. (2005). Sensitivity and convergence of uniformly ergodic Markov chains. To appear in J. Appl. Prob. CrossRefGoogle Scholar
Peshkin, L. and Gelfand, M. S. (1999). Segmentation of yeast DNA using hidden Markov models. Bioinformatics 15, 980–986.CrossRefGoogle ScholarPubMed
Petrie, T. (1969). Probabilistic functions of finite state Markov chains. Ann. Math. Statist. 40, 97–115.CrossRefGoogle Scholar
Pyke, R. (1961). Markov renewal processes: definitions and preliminary properties. Ann. Math. Statist. 32, 1231–1242.CrossRefGoogle Scholar
Qin, F., Auerbach, A. and Sachs, F. (2000). A direct optimization approach for hidden Markov modelling for single channel kinetics. Biophys. J. 79, 1915–1915.CrossRefGoogle Scholar
Rabiner, L. R. (1989). A tutorial on hidden Markov models and selected applications in speech recognition. Proc. IEEE 77, 257–284.CrossRefGoogle Scholar
Rosales, R., Stark, J. A., Fitzgerald, W. J. and Hladky, S. B. (2001). Bayesian restoration of ion channel recordings using hidden Markov models. Biophys. J. 80, 1088–1103.CrossRefGoogle ScholarPubMed
Seneta, E. (1981). Nonnegative Matrices and Markov Chains. Springer, New York.CrossRefGoogle Scholar
Seneta, E. (1984). Explicit forms for ergodicity coefficients and spectrum localization. Linear Algebra Appl. 60, 187–197.CrossRefGoogle Scholar
Zeifman, A. I. (1995). Upper and lower bounds on the rate of convergence for nonhomogeneous birth and death processes. Stoch. Process. Appl. 59, 157–173.CrossRefGoogle Scholar
Zeifman, A. I. and Isaacson, D. L. (1994). On strong ergodicity for nonhomogeneous continuous-time Markov chains. Stoch. Process. Appl. 50, 263–273.CrossRefGoogle Scholar