Hidden Markov model: Difference between revisions

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Finally, a different rationale towards addressing the problem of modeling nonstationary data by means of hidden Markov models was suggested in 2012.<ref name="Reservoir-HMM">{{cite journal |last1=Chatzis |first1=Sotirios P. |last2=Demiris |first2=Yiannis |year=2012 |title=A Reservoir-Driven Non-Stationary Hidden Markov Model |journal=Pattern Recognition |volume=45 |issue=11 |pages=3985–3996 |doi=10.1016/j.patcog.2012.04.018|bibcode=2012PatRe..45.3985C |hdl=10044/1/12611 |hdl-access=free }}</ref> It consists in employing a small recurrent neural network (RNN), specifically a reservoir network,<ref>M. Lukosevicius, H. Jaeger (2009) Reservoir computing approaches to recurrent neural network training, Computer Science Review '''3''': 127–149.</ref> to capture the evolution of the temporal dynamics in the observed data. This information, encoded in the form of a high-dimensional vector, is used as a conditioning variable of the HMM state transition probabilities. Under such a setup, we eventually obtain a nonstationary HMM the transition probabilities of which evolve over time in a manner that is inferred from the data itself, as opposed to some unrealistic ad-hoc model of temporal evolution.

In 2023, two innovative algorithms were introduced for the Hidden Markov Model. These algorithms enable the computation of the posterior distribution of the HMM without the necessity of explicitly modeling the joint distribution, utilizing only the conditional distributions .<ref>Azeraf, E., Monfrini, E., & Pieczynski, W. (2023). Equivalence between LC-CRF and HMM, and Discriminative Computing of HMM-Based MPM and MAP. Algorithms, 16(3), 173.</ref><ref>Azeraf, E., Monfrini, E., Vignon, E., & Pieczynski, W. (2020). Hidden markov chains, entropic forward-backward, and part-of-speech tagging. arXiv preprint arXiv:2005.10629.</ref>. Unlike traditional methods such as the Forward-Backward and Viterbi algorithms, which require knowledge of the joint law of the HMM and can be computationally intensive to learn, the Discriminative Forward-Backward and Discriminative Viterbi algorithms circumvent the need for the observation's law .<ref>Azeraf, E., Monfrini, E., & Pieczynski, W. (2022). Deriving discriminative classifiers from generative models. arXiv preprint arXiv:2201.00844.</ref><ref>Ng, A., & Jordan, M. (2001). On discriminative vs. generative classifiers: A comparison of logistic regression and naive bayes. Advances in neural information processing systems, 14.</ref>. This breakthrough allows the HMM to be applied as a discriminative model, offering a more efficient and versatile approach to leveraging Hidden Markov Models in various applications.

The model suitable in the context of longitudinal data is named latent Markov model.<ref>{{Cite book|title=Panel Analysis: Latent Probability Models for Attitude and Behaviour Processes|last=Wiggins|first=L. M.|publisher=Elsevier|year=1973|___location=Amsterdam}}</ref> The basic version of this model has been extended to include individual covariates, random effects and to model more complex data structures such as multilevel data. A complete overview of the latent Markov models, with special attention to the model assumptions and to their practical use is provided in<ref>{{Cite book|url=https://sites.google.com/site/latentmarkovbook/home|title=Latent Markov models for longitudinal data|last1=Bartolucci|first1=F.|last2=Farcomeni|first2=A.|last3=Pennoni|first3=F.|publisher=Chapman and Hall/CRC|year=2013|isbn=978-14-3981-708-7|___location=Boca Raton}}</ref>