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The parameter learning task in HMMs is to find, given an output sequence or a set of such sequences, the best set of state transition and emission probabilities.

The task is usually to derive the maximum likelihood estimate of the parameters of the HMM given the set of output sequences.

No tractable algorithm is known for solving this problem exactly, but a local maximum likelihood can be derived efficiently using the Baum—Welch algorithm or the Baldi—Chauvin algorithm. The Baum—Welch algorithm is a special case of the expectation-maximization algorithm. If the HMMs are used for time series prediction, more sophisticated Bayesian inference methods, like Markov chain Monte Carlo MCMC sampling are proven to be favorable over finding a single maximum likelihood model both in terms of accuracy and stability.

Typical learning models correspond to assuming a discrete uniform distribution over possible states i. As mentioned above, the distribution of each observation in a hidden Markov model is a mixture density , with the states of the corresponding to mixture components. It is useful to compare the above characterizations for an HMM with the corresponding characterizations, of a mixture model , using the same notation.

The following mathematical descriptions are fully written out and explained, for ease of implementation. In other words, the path followed by the Markov chain of hidden states will be highly random.

An alternative for the above two Bayesian examples would be to add another level of prior parameters for the transition matrix. That is, replace the lines. Hence, a two-level model such as just described allows independent control over 1 the overall density of the transition matrix, and 2 the density of states to which transitions are likely i.

In both cases this is done while still assuming ignorance over which particular states are more likely than others. Poisson hidden Markov models PHMM are special cases of hidden Markov models where a Poisson process has a rate which varies in association with changes between the different states of a Markov model. HMMs can be applied in many fields where the goal is to recover a data sequence that is not immediately observable but other data that depend on the sequence are.

The forward and backward recursions used in HMM as well as computations of marginal smoothing probabilities were first described by Ruslan L. Stratonovich in [6] pages — and in the late s in his papers in Russian. Baum and other authors in the second half of the s. One of the first applications of HMMs was speech recognition , starting in the mids. In the second half of the s, HMMs began to be applied to the analysis of biological sequences, [37] in particular DNA.

Since then, they have become ubiquitous in the field of bioinformatics. Hidden Markov models can model complex Markov processes where the states emit the observations according to some probability distribution. One such example is the Gaussian distribution; in such a Hidden Markov Model the states output are represented by a Gaussian distribution. Moreover, it could represent even more complex behavior when the output of the states is represented as mixture of two or more Gaussians, in which case the probability of generating an observation is the product of the probability of first selecting one of the Gaussians and the probability of generating that observation from that Gaussian.

In cases of modeled data exhibiting artifacts such as outliers and skewness, one may resort to finite mixtures of heavier-tailed elliptical distributions, such as the multivariate Student's-t distribution, or appropriate non-elliptical distributions, such as the multivariate Normal Inverse-Gaussian. 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.

Hidden Markov models are generative models , 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 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 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 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. It was originally described under the name "Infinite Hidden Markov Model" [4] and was further formalized in [5].

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 model".

The advantage of this type of model is that arbitrary features i. 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: 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. In practice, approximate techniques, such as variational approaches, could be used. All of the above models can be extended to allow for more distant dependencies among hidden states, e.

Another recent extension is the triplet Markov model , [41] 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 [11] and which allows to fuse data in Markovian context [12] and to model nonstationary data.

Finally, a different rationale towards addressing the problem of modeling nonstationary data by means of hidden Markov models was suggested in 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.

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 [46]. From Wikipedia, the free encyclopedia. Machine learning and data mining Problems. Graphical models Bayes net Conditional random field Hidden Markov.

Glossary of artificial intelligence. List of datasets for machine-learning research Outline of machine learning. The Annals of Mathematical Statistics. Retrieved 28 November Bulletin of the American Mathematical Society. Pacific Journal of Mathematics. Theory of Probability and its Applications.

International Journal of Approximate Reasoning. Proceedings of the IEEE. Expert Systems with Applications. Biophysical Reviews and Letters. Journal in Computer Virology.

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