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Regenerativity of Viterbi Process for Pairwise Markov Models

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  • Jüri Lember
  • Joonas Sova

Abstract

For hidden Markov models, one of the most popular estimates of the hidden chain is the Viterbi path—the path maximising the posterior probability. We consider a more general setting, called the pairwise Markov model (PMM), where the joint process consisting of finite-state hidden process and observation process is assumed to be a Markov chain. It has been recently proven that under some conditions the Viterbi path of the PMM can almost surely be extended to infinity, thereby defining the infinite Viterbi decoding of the observation sequence, called the Viterbi process. This was done by constructing a block of observations, called a barrier, which ensures that the Viterbi path goes through a given state whenever this block occurs in the observation sequence. In this paper, we prove that the joint process consisting of Viterbi process and PMM is regenerative. The proof involves a delicate construction of regeneration times which coincide with the occurrences of barriers. As one possible application of our theory, some results on the asymptotics of the Viterbi training algorithm are derived.

Suggested Citation

  • Jüri Lember & Joonas Sova, 2021. "Regenerativity of Viterbi Process for Pairwise Markov Models," Journal of Theoretical Probability, Springer, vol. 34(1), pages 1-33, March.
    • Handle: RePEc:spr:jotpro:v:34:y:2021:i:1:d:10.1007_s10959-020-01022-z
    DOI: 10.1007/s10959-020-01022-z
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    References listed on IDEAS

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    1. Vladimir V. Kalashnikov, 1994. "Topics on regenerative processes," International Journal of Stochastic Analysis, Hindawi, vol. 7, pages 1-1, January.
    2. Kristi Kuljus & Jüri Lember, 2016. "On the Accuracy of the MAP Inference in HMMs," Methodology and Computing in Applied Probability, Springer, vol. 18(3), pages 597-627, September.
    3. Hamilton, James D, 1989. "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle," Econometrica, Econometric Society, vol. 57(2), pages 357-384, March.
    4. Lember, Jüri & Sova, Joonas, 2020. "Existence of infinite Viterbi path for pairwise Markov models," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1388-1425.
    5. Lember, Jüri, 2011. "A correction on approximation of smoothing probabilities for hidden Markov models," Statistics & Probability Letters, Elsevier, vol. 81(9), pages 1463-1464, September.
    6. Derrode, Stéphane & Pieczynski, Wojciech, 2013. "Unsupervised data classification using pairwise Markov chains with automatic copulas selection," Computational Statistics & Data Analysis, Elsevier, vol. 63(C), pages 81-98.
    7. Lember, Jüri, 2011. "On approximation of smoothing probabilities for hidden Markov models," Statistics & Probability Letters, Elsevier, vol. 81(2), pages 310-316, February.
    8. Hamilton, James D., 1990. "Analysis of time series subject to changes in regime," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 39-70.
    9. Celeux, Gilles & Govaert, Gerard, 1992. "A classification EM algorithm for clustering and two stochastic versions," Computational Statistics & Data Analysis, Elsevier, vol. 14(3), pages 315-332, October.
    10. Goodwin, Thomas H, 1993. "Business-Cycle Analysis with a Markov-Switching Model," Journal of Business & Economic Statistics, American Statistical Association, vol. 11(3), pages 331-339, July.
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