Document Open Access Logo

On the Sequential Probability Ratio Test in Hidden Markov Models

Authors Oscar Darwin , Stefan Kiefer

Thumbnail PDF


  • Filesize: 0.83 MB
  • 16 pages

Document Identifiers

Author Details

Oscar Darwin
  • Department of Computer Science, Oxford University, UK
Stefan Kiefer
  • Department of Computer Science, Oxford University, UK


The authors thank anonymous referees for valuable suggestions.

Cite AsGet BibTex

Oscar Darwin and Stefan Kiefer. On the Sequential Probability Ratio Test in Hidden Markov Models. In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 9:1-9:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


We consider the Sequential Probability Ratio Test applied to Hidden Markov Models. Given two Hidden Markov Models and a sequence of observations generated by one of them, the Sequential Probability Ratio Test attempts to decide which model produced the sequence. We show relationships between the execution time of such an algorithm and Lyapunov exponents of random matrix systems. Further, we give complexity results about the execution time taken by the Sequential Probability Ratio Test.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
  • Mathematics of computing → Stochastic processes
  • Theory of computation → Logic and verification
  • Markov chains
  • hidden Markov models
  • probabilistic systems
  • verification


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. P. Ailliot, C. Thompson, and P. Thomson. Space-time modelling of precipitation by using a hidden Markov model and censored Gaussian distributions. Journal of the Royal Statistical Society, 58(3):405-426, 2009. Google Scholar
  2. S. Akshay, H. Bazille, E. Fabre, and B. Genest. Classification among hidden Markov models. In Proceedings of the Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), volume 150 of LIPIcs, pages 29:1-29:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL:
  3. M. Alexandersson, S. Cawley, and L. Pachter. SLAM: Cross-species gene finding and alignment with a generalized pair hidden Markov model. Genome Research, 13:469-502, 2003. Google Scholar
  4. N. Bertrand, S. Haddad, and E. Lefaucheux. Accurate approximate diagnosability of stochastic systems. In Proceedings of Language and Automata Theory and Applications (LATA), volume 9618 of Lecture Notes in Computer Science, pages 549-561. Springer, 2016. URL:
  5. B. Chen and P. Willett. Detection of hidden Markov model transient signals. IEEE Transactions on Aerospace and Electronic Systems, 36(4):1253-1268, 2000. URL:
  6. F.-S. Chen, C.-M. Fu, and C.-L. Huang. Hand gesture recognition using a real-time tracking method and hidden Markov models. Image and Vision Computing, 21(8):745-758, 2003. Google Scholar
  7. T. Chen and S. Kiefer. On the total variation distance of labelled Markov chains. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pages 33:1-33:10, Vienna, Austria, 2014. Google Scholar
  8. G.A. Churchill. Stochastic models for heterogeneous DNA sequences. Bulletin of Mathematical Biology, 51(1):79-94, 1989. Google Scholar
  9. C. Cortes, M. Mohri, and A. Rastogi. L_p distance and equivalence of probabilistic automata. International Journal of Foundations of Computer Science, 18(04):761-779, 2007. Google Scholar
  10. M.S. Crouse, R.D. Nowak, and R.G. Baraniuk. Wavelet-based statistical signal processing using hidden Markov models. IEEE Transactions on Signal Processing, 46(4):886-902, April 1998. Google Scholar
  11. O. Darwin and S. Kiefer. On the sequential probability ratio test in hidden Markov models, 2022. URL:
  12. C. Dehnert, S. Junges, J.-P. Katoen, and M. Volk. A Storm is coming: A modern probabilistic model checker. In Proceedings of Computer Aided Verification (CAV), pages 592-600. Springer, 2017. Google Scholar
  13. R. Durbin. Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids. Cambridge University Press, 1998. Google Scholar
  14. S.R. Eddy. What is a hidden Markov model? Nature Biotechnology, 22(10):1315-1316, October 2004. Google Scholar
  15. K. Etessami, A. Stewart, and M. Yannakakis. A note on the complexity of comparing succinctly represented integers, with an application to maximum probability parsing. ACM Trans. Comput. Theory, 6(2):9:1-9:23, 2014. URL:
  16. C.-D. Fuh. SPRT and CUSUM in hidden Markov models. The Annals of Statistics, 31(3):942-977, 2003. URL:
  17. E. Grossi and M. Lops. Sequential detection of Markov targets with trajectory estimation. IEEE Transactions on Information Theory, 54(9):4144-4154, 2008. URL:
  18. B.-H. Juang and L. R. Rabiner. A probabilistic distance measure for hidden Markov models. AT&T Technical Journal, 64(2):391-408, 1985. URL:
  19. J.-Y. Kao, N. Rampersad, and J. Shallit. On NFAs where all states are final, initial, or both. Theoretical Computer Science, 410(47):5010-5021, 2009. URL:
  20. S. Kiefer, A.S. Murawski, J. Ouaknine, B. Wachter, and J. Worrell. Language equivalence for probabilistic automata. In Proceedings of the 23rd International Conference on Computer Aided Verification (CAV), volume 6806 of LNCS, pages 526-540. Springer, 2011. Google Scholar
  21. S. Kiefer and A.P. Sistla. Distinguishing hidden Markov chains. In Proceedings of the 31st Annual Symposium on Logic in Computer Science (LICS), pages 66-75, New York, USA, 2016. ACM. Google Scholar
  22. A. Krogh, B. Larsson, G. von Heijne, and E.L.L. Sonnhammer. Predicting transmembrane protein topology with a hidden Markov model: Application to complete genomes. Journal of Molecular Biology, 305(3):567-580, 2001. Google Scholar
  23. M. Kwiatkowska, G. Norman, and D. Parker. PRISM 4.0: Verification of probabilistic real-time systems. In Proceedings of Computer Aided Verification (CAV), volume 6806 of LNCS, pages 585-591. Springer, 2011. Google Scholar
  24. R. Langrock, B. Swihart, B. Caffo, N. Punjabi, and C. Crainiceanu. Combining hidden Markov models for comparing the dynamics of multiple sleep electroencephalograms. Statistics in medicine, 32, August 2013. URL:
  25. A. Paz. Introduction to Probabilistic Automata (Computer Science and Applied Mathematics). Academic Press, Inc., Orlando, FL, USA, 1971. Google Scholar
  26. V.Yu. Protasov. Asymptotics of products of nonnegative random matrices. Functional Analysis and Its Applications, 47:138-147, 2013. Google Scholar
  27. V.Yu. Protasov and R.M. Jungers. Lower and upper bounds for the largest Lyapunov exponent of matrices. Linear Algebra and its Applications, 438(11):4448-4468, 2013. URL:
  28. L.R. Rabiner. A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of the IEEE, 77(2):257-286, 1989. Google Scholar
  29. M.P. Schützenberger. On the definition of a family of automata. Information and Control, 4(2):245-270, 1961. Google Scholar
  30. D. Sutter, O. Fawzi, and R. Renner. Bounds on Lyapunov exponents via entropy accumulation. IEEE Transactions on Information Theory, 67(1):10-24, 2021. URL:
  31. W.-G. Tzeng. A polynomial-time algorithm for the equivalence of probabilistic automata. SIAM J. Comput., 21(2):216-227, April 1992. Google Scholar
  32. A. Wald. Sequential Tests of Statistical Hypotheses. The Annals of Mathematical Statistics, 16(2):117-186, 1945. URL:
  33. A. Wald and J. Wolfowitz. Optimum character of the sequential probability ratio test. The Annals of Mathematical Statistics, 19(3):326-339, 1948. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail