Editing Hidden Markov model (section)

== Extensions ==
=== General state spaces ===
In the hidden Markov models considered above, the state space of the hidden variables is discrete, while the observations themselves can either be discrete (typically generated from a [[categorical distribution]]) or continuous (typically from a [[Gaussian distribution]]). Hidden Markov models can also be generalized to allow continuous state spaces. Examples of such models are those where the Markov process over hidden variables is a [[linear dynamical system]], with a linear relationship among related variables and where all hidden and observed variables follow a [[Gaussian distribution]]. In simple cases, such as the linear dynamical system just mentioned, exact inference is tractable (in this case, using the [[Kalman filter]]); however, in general, exact inference in HMMs with continuous latent variables is infeasible, and approximate methods must be used, such as the [[extended Kalman filter]] or the [[particle filter]].

Nowadays, inference in hidden Markov models is performed in [[Nonparametric statistics|nonparametric]] settings, where the dependency structure enables  [[identifiability]] of the model<ref>{{Cite journal |last1=Gassiat |first1=E. |last2=Cleynen |first2=A. |last3=Robin |first3=S. |date=2016-01-01 |title=Inference in finite state space non parametric Hidden Markov Models and applications |url=https://doi.org/10.1007/s11222-014-9523-8 |journal=Statistics and Computing |language=en |volume=26 |issue=1 |pages=61–71 |doi=10.1007/s11222-014-9523-8 |issn=1573-1375|url-access=subscription }}</ref> and the learnability limits are still under exploration.<ref>{{Cite journal |last1=Abraham |first1=Kweku |last2=Gassiat |first2=Elisabeth |last3=Naulet |first3=Zacharie |date=March 2023 |title=Fundamental Limits for Learning Hidden Markov Model Parameters |url=https://ieeexplore.ieee.org/document/9917566 |journal=IEEE Transactions on Information Theory |volume=69 |issue=3 |pages=1777–1794 |doi=10.1109/TIT.2022.3213429 |arxiv=2106.12936 |issn=0018-9448}}</ref>

=== Bayesian modeling of the transitions probabilities ===
Hidden Markov models are [[generative model]]s, in which the [[joint distribution]] of observations and hidden states, or equivalently both the [[prior distribution]] of hidden states (the ''transition probabilities'') and [[conditional distribution]] of observations given states (the ''emission probabilities''), is modeled. The above algorithms implicitly assume a [[Uniform distribution (continuous)|uniform]] prior distribution over the transition probabilities. However, it is also possible to create hidden Markov models with other types of prior distributions. An obvious candidate, given the categorical distribution of the transition probabilities, is the [[Dirichlet distribution]], which is the [[conjugate prior]] distribution of the categorical distribution. Typically, a symmetric Dirichlet distribution is chosen, reflecting ignorance about which states are inherently more likely than others. The single parameter of this distribution (termed the ''concentration parameter'') controls the relative density or sparseness of the resulting transition matrix. A choice of 1 yields a uniform distribution. Values greater than 1 produce a dense matrix, in which the transition probabilities between pairs of states are likely to be nearly equal. Values less than 1 result in a sparse matrix in which, for each given source state, only a small number of destination states have non-negligible transition probabilities. It is also possible to use a two-level prior Dirichlet distribution, in which one Dirichlet distribution (the upper distribution) governs the parameters of another Dirichlet distribution (the lower distribution), which in turn governs the transition probabilities. The upper distribution governs the overall distribution of states, determining how likely each state is to occur; its concentration parameter determines the density or sparseness of states. Such a two-level prior distribution, where both concentration parameters are set to produce sparse distributions, might be useful for example in [[unsupervised learning|unsupervised]] [[part-of-speech tagging]], where some parts of speech occur much more commonly than others; learning algorithms that assume a uniform prior distribution generally perform poorly on this task. The parameters of models of this sort, with non-uniform prior distributions, can be learned using [[Gibbs sampling]] or extended versions of the [[expectation-maximization algorithm]].

An extension of the previously described hidden Markov models with [[Dirichlet distribution|Dirichlet]] priors uses a [[Dirichlet process]] in place of a Dirichlet distribution. This type of model allows for an unknown and potentially infinite number of states. It is common to use a two-level Dirichlet process, similar to the previously described model with two levels of Dirichlet distributions. Such a model is called a ''hierarchical Dirichlet process hidden Markov model'', or ''HDP-HMM'' for short. It was originally described under the name "Infinite Hidden Markov Model"<ref>Beal, Matthew J., Zoubin Ghahramani, and Carl Edward Rasmussen. "The infinite hidden Markov model." Advances in neural information processing systems 14 (2002): 577-584.</ref> and was further formalized in "Hierarchical Dirichlet Processes".<ref>Teh, Yee Whye, et al. "Hierarchical dirichlet processes." Journal of the American Statistical Association 101.476 (2006).</ref>

=== Discriminative approach ===
A different type of extension uses a [[discriminative model]] in place of the [[generative model]] of standard HMMs. This type of model directly models the conditional distribution of the hidden states given the observations, rather than modeling the joint distribution. An example of this model is the so-called ''[[maximum entropy Markov model]]'' (MEMM), which models the conditional distribution of the states using [[logistic regression]] (also known as a "[[Maximum entropy probability distribution|maximum entropy]] model"). The advantage of this type of model is that arbitrary features (i.e. functions) of the observations can be modeled, allowing domain-specific knowledge of the problem at hand to be injected into the model. Models of this sort are not limited to modeling direct dependencies between a hidden state and its associated observation; rather, features of nearby observations, of combinations of the associated observation and nearby observations, or in fact of arbitrary observations at any distance from a given hidden state can be included in the process used to determine the value of a hidden state. Furthermore, there is no need for these features to be [[statistically independent]] of each other, as would be the case if such features were used in a generative model. Finally, arbitrary features over pairs of adjacent hidden states can be used rather than simple transition probabilities. The disadvantages of such models are: (1) The types of prior distributions that can be placed on hidden states are severely limited; (2) It is not possible to predict the probability of seeing an arbitrary observation. This second limitation is often not an issue in practice, since many common usages of HMM's do not require such predictive probabilities.

A variant of the previously described discriminative model is the linear-chain [[conditional random field]]. This uses an undirected graphical model (aka [[Markov random field]]) rather than the directed graphical models of MEMM's and similar models. The advantage of this type of model is that it does not suffer from the so-called ''label bias'' problem of MEMM's, and thus may make more accurate predictions. The disadvantage is that training can be slower than for MEMM's.

=== Other extensions ===
Yet another variant is the ''factorial hidden Markov model'', which allows for a single observation to be conditioned on the corresponding hidden variables of a set of <math>K</math> independent Markov chains, rather than a single Markov chain. It is equivalent to a single HMM, with <math>N^K</math> states (assuming there are <math>N</math> states for each chain), and therefore, learning in such a model is difficult: for a sequence of length <math>T</math>, a straightforward Viterbi algorithm has complexity <math>O(N^{2K} \, T)</math>. To find an exact solution, a junction tree algorithm could be used, but it results in an <math>O(N^{K+1} \, K \, T)</math> complexity. In practice, approximate techniques, such as variational approaches, could be used.<ref>{{cite journal |last1=Ghahramani |first1=Zoubin |author-link1=Zoubin Ghahramani |last2=Jordan |first2=Michael I. |author-link2=Michael I. Jordan |title=Factorial Hidden Markov Models |journal=[[Machine Learning (journal)|Machine Learning]] |year=1997 |volume=29 |issue=2/3 |pages=245–273 |doi=10.1023/A:1007425814087|doi-access=free}}</ref>

All of the above models can be extended to allow for more distant dependencies among hidden states, e.g. allowing for a given state to be dependent on the previous two or three states rather than a single previous state; i.e. the transition probabilities are extended to encompass sets of three or four adjacent states (or in general <math>K</math> adjacent states). The disadvantage of such models is that dynamic-programming algorithms for training them have an <math>O(N^K \, T)</math> running time, for <math>K</math> adjacent states and <math>T</math> total observations (i.e. a length-<math>T</math> Markov chain). This extension has been widely used in [[bioinformatics]], in the modeling of [[Nucleic acid sequence|DNA sequences]].

Another recent extension is the ''triplet Markov model'',<ref name="TMM">{{cite journal |doi=10.1016/S1631-073X(02)02462-7 |volume=335 |issue=3 |title=Chaı̂nes de Markov Triplet |year=2002 |journal=Comptes Rendus Mathématique |pages=275–278 |last1=Pieczynski |first1=Wojciech|url=http://www.numdam.org/item/10.1016/S1631-073X(02)02462-7.pdf}}</ref> in which an auxiliary underlying process is added to model some data specificities. Many variants of this model have been proposed. One should also mention the interesting link that has been established between the ''theory of evidence'' and the ''triplet Markov models''<ref name="TMMEV">{{cite journal |doi=10.1016/j.ijar.2006.05.001 |volume=45 |title=Multisensor triplet Markov chains and theory of evidence |year=2007 |journal=International Journal of Approximate Reasoning |pages=1–16 |last1=Pieczynski |first1=Wojciech|doi-access=free}}</ref> and which allows to fuse data in Markovian context<ref name="JASP">[http://asp.eurasipjournals.com/content/pdf/1687-6180-2012-134.pdf Boudaren et al.] {{Webarchive|url=https://web.archive.org/web/20140311164443/http://asp.eurasipjournals.com/content/pdf/1687-6180-2012-134.pdf |date=2014-03-11}}, M. Y. Boudaren, E. Monfrini, W. Pieczynski, and A. Aissani, Dempster-Shafer fusion of multisensor signals in nonstationary Markovian context, EURASIP Journal on Advances in Signal Processing, No. 134, 2012.</ref> and to model nonstationary data.<ref name="TSP">[https://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=1468502&contentType=Journals+%26+Magazines&searchField%3DSearch_All%26queryText%3Dlanchantin+pieczynski Lanchantin et al.], P. Lanchantin and W. Pieczynski, Unsupervised restoration of hidden non stationary Markov chain using evidential priors, IEEE Transactions on Signal Processing, Vol. 53, No. 8, pp. 3091-3098, 2005.</ref><ref name="SPL">[https://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=6244854&contentType=Journals+%26+Magazines&searchField%3DSearch_All%26queryText%3Dboudaren Boudaren et al.], M. Y. Boudaren, E. Monfrini, and W. Pieczynski, Unsupervised segmentation of random discrete data hidden with switching noise distributions, IEEE Signal Processing Letters, Vol. 19, No. 10, pp. 619-622, October 2012.</ref> Alternative multi-stream data fusion strategies have also been proposed in recent literature, e.g.,<ref>Sotirios P. Chatzis, Dimitrios Kosmopoulos, [https://ieeexplore.ieee.org/document/6164251/ "Visual Workflow Recognition Using a Variational Bayesian Treatment of Multistream Fused Hidden Markov Models,"] IEEE Transactions on Circuits and Systems for Video Technology, vol. 22, no. 7, pp. 1076-1086, July 2012.</ref>

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, eventually is obtained a nonstationary HMM, the transition probabilities of which evolve over time in a manner that is inferred from the data, in contrast 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>