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121-480: A hidden Markov model ( HMM ) is a Markov model in which the observations are dependent on a latent (or hidden ) Markov process (referred to as X {\displaystyle X} ). An HMM requires that there be an observable process Y {\displaystyle Y} whose outcomes depend on the outcomes of X {\displaystyle X} in a known way. Since X {\displaystyle X} cannot be observed directly,

242-482: 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

363-453: A Gaussian distribution ). The parameters of a hidden Markov model are of two types, transition probabilities and emission probabilities (also known as output probabilities ). The transition probabilities control the way the hidden state at time t is chosen given the hidden state at time t − 1 {\displaystyle t-1} . The hidden state space is assumed to consist of one of N possible values, modelled as

484-413: 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

605-575: A log-normal distribution . The density of Y follows with f X {\displaystyle f_{X}} standard Normal and g − 1 ( y ) = log ⁡ ( y ) {\displaystyle g^{-1}(y)=\log(y)} , | ( g − 1 ( y ) ) ′ | = 1 y {\displaystyle |(g^{-1}(y))^{\prime }|={\frac {1}{y}}} for y > 0 {\displaystyle y>0} . As assumed above, if

726-401: A parametric family { f ( ⋅ ; θ ) ∣ θ ∈ Θ } , {\displaystyle \;\{f(\cdot \,;\theta )\mid \theta \in \Theta \}\;,} where Θ {\displaystyle \,\Theta \,} is called the parameter space , a finite-dimensional subset of Euclidean space . Evaluating

847-463: 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

968-430: A vector-valued function mapping R k {\displaystyle \,\mathbb {R} ^{k}\,} into R r   . {\displaystyle \;\mathbb {R} ^{r}~.} Estimating the true parameter θ {\displaystyle \theta } belonging to Θ {\displaystyle \Theta } then, as a practical matter, means to find

1089-417: A " 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

1210-454: A 30% chance that tomorrow will be sunny if today is rainy. The emission_probability represents how likely Bob is to perform a certain activity on each day. If it is rainy, there is a 50% chance that he is cleaning his apartment; if it is sunny, there is a 60% chance that he is outside for a walk. A similar example is further elaborated in the Viterbi algorithm page. The diagram below shows

1331-561: A categorical distribution. (See the section below on extensions for other possibilities.) This means that for each of the N possible states that a hidden variable at time t can be in, there is a transition probability from this state to each of the N possible states of the hidden variable at time t + 1 {\displaystyle t+1} , for a total of N 2 {\displaystyle N^{2}} transition probabilities. The set of transition probabilities for transitions from any given state must sum to 1. Thus,

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1452-473: A classifier that minimizes total expected risk, especially, when the costs (the loss function) associated with different decisions are equal, the classifier is minimizing the error over the whole distribution. Thus, the Bayes Decision Rule is stated as where w 1 , w 2 {\displaystyle \;w_{1}\,,w_{2}\;} are predictions of different classes. From

1573-416: A different rationale towards addressing the problem of modeling nonstationary data by means of hidden Markov models was suggested in 2012. It consists in employing a small recurrent neural network (RNN), specifically a reservoir network, 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

1694-526: A factor that does not depend on the model parameters. For example, the MLE parameters of the log-normal distribution are the same as those of the normal distribution fitted to the logarithm of the data. In fact, in the log-normal case if X ∼ N ( 0 , 1 ) {\displaystyle X\sim {\mathcal {N}}(0,1)} , then Y = g ( X ) = e X {\displaystyle Y=g(X)=e^{X}} follows

1815-474: A forecasting methods for several topics, for example price trends, wind power and solar irradiance . The Markov-chain forecasting models utilize a variety of different settings, from discretizing the time-series to hidden Markov-models combined with wavelets and the Markov-chain mixture distribution model (MCM). Maximum likelihood estimation In statistics , maximum likelihood estimation ( MLE )

1936-418: A known mix of balls, with each ball having a unique label y1, y2, y3, ... . The genie chooses an urn in that room and randomly draws a ball from that urn. It then puts the ball onto a conveyor belt, where the observer can observe the sequence of the balls but not the sequence of urns from which they were drawn. The genie has some procedure to choose urns; the choice of the urn for the n -th ball depends only upon

2057-771: 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. Since MCMC imposes significant computational burden, in cases where computational scalability

2178-406: A particular sequence of observations (see illustration on the right). This task is generally applicable when HMM's are applied to different sorts of problems from those for which the tasks of filtering and smoothing are applicable. An example is part-of-speech tagging , where the hidden states represent the underlying parts of speech corresponding to an observed sequence of words. In this case, what

2299-532: A person's location in a room, can be interpreted to determine more complex information, such as in what task or activity the person is performing. Two kinds of Hierarchical Markov Models are the Hierarchical hidden Markov model and the Abstract Hidden Markov Model. Both have been used for behavior recognition and certain conditional independence properties between different levels of abstraction in

2420-455: A probability measure on the subshifts on A , B {\displaystyle A,B} . Markov model There are four common Markov models used in different situations, depending on whether every sequential state is observable or not, and whether the system is to be adjusted on the basis of observations made: The simplest Markov model is the Markov chain . It models the state of

2541-403: A random number and the choice of the urn for the ( n − 1)-th ball. The choice of urn does not directly depend on the urns chosen before this single previous urn; therefore, this is called a Markov process . It can be described by the upper part of Figure 1. The Markov process cannot be observed, only the sequence of labeled balls, thus this arrangement is called a hidden Markov process . This

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2662-498: A sequence drawn from some null distribution will have an HMM probability (in the case of the forward algorithm) or a maximum state sequence probability (in the case of the Viterbi algorithm) at least as large as that of a particular output sequence? When an HMM is used to evaluate the relevance of a hypothesis for a particular output sequence, the statistical significance indicates the false positive rate associated with failing to reject

2783-523: A sequence of length T {\displaystyle T} , a straightforward Viterbi algorithm has complexity O ( N 2 K T ) {\displaystyle O(N^{2K}\,T)} . To find an exact solution, a junction tree algorithm could be used, but it results in an O ( N K + 1 K T ) {\displaystyle O(N^{K+1}\,K\,T)} complexity. In practice, approximate techniques, such as variational approaches, could be used. All of

2904-494: A set h 1 , h 2 , … , h r , h r + 1 , … , h k {\displaystyle \;h_{1},h_{2},\ldots ,h_{r},h_{r+1},\ldots ,h_{k}\;} in such a way that h ∗ = [ h 1 , h 2 , … , h k ] {\displaystyle \;h^{\ast }=\left[h_{1},h_{2},\ldots ,h_{k}\right]\;}

3025-449: A set of emission probabilities governing the distribution of the observed variable at a particular time given the state of the hidden variable at that time. The size of this set depends on the nature of the observed variable. For example, if the observed variable is discrete with M possible values, governed by a categorical distribution , there will be M − 1 {\displaystyle M-1} separate parameters, for

3146-438: 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

3267-530: A statistical test of the "validity" of the constraint, known as the Lagrange multiplier test . Nonparametric maximum likelihood estimation can be performed using the empirical likelihood . A maximum likelihood estimator is an extremum estimator obtained by maximizing, as a function of θ , the objective function ℓ ^ ( θ ; x ) {\displaystyle {\widehat {\ell \,}}(\theta \,;x)} . If

3388-439: A stronger condition of uniform convergence almost surely has to be imposed: Additionally, if (as assumed above) the data were generated by f ( ⋅ ; θ 0 ) {\displaystyle f(\cdot \,;\theta _{0})} , then under certain conditions, it can also be shown that the maximum likelihood estimator converges in distribution to a normal distribution. Specifically, where I

3509-429: A system with a random variable that changes through time. In this context, the Markov property indicates that the distribution for this variable depends only on the distribution of a previous state. An example use of a Markov chain is Markov chain Monte Carlo , which uses the Markov property to prove that a particular method for performing a random walk will sample from the joint distribution . A hidden Markov model

3630-473: A total of N ( M − 1 ) {\displaystyle N(M-1)} emission parameters over all hidden states. On the other hand, if the observed variable is an M -dimensional vector distributed according to an arbitrary multivariate Gaussian distribution , there will be M parameters controlling the means and M ( M + 1 ) 2 {\displaystyle {\frac {M(M+1)}{2}}} parameters controlling

3751-488: 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

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3872-418: A unique global maximum. Compactness implies that the likelihood cannot approach the maximum value arbitrarily close at some other point (as demonstrated for example in the picture on the right). Compactness is only a sufficient condition and not a necessary condition. Compactness can be replaced by some other conditions, such as: The dominance condition can be employed in the case of i.i.d. observations. In

3993-500: Is negative semi-definite at θ ^ {\displaystyle {\widehat {\theta \,}}} , as this indicates local concavity . Conveniently, most common probability distributions – in particular the exponential family – are logarithmically concave . While the domain of the likelihood function—the parameter space —is generally a finite-dimensional subset of Euclidean space , additional restrictions sometimes need to be incorporated into

4114-1209: Is a hidden Markov model if Let X t {\displaystyle X_{t}} and Y t {\displaystyle Y_{t}} be continuous-time stochastic processes. The pair ( X t , Y t ) {\displaystyle (X_{t},Y_{t})} is a hidden Markov model if The states of the process X n {\displaystyle X_{n}} (resp. X t ) {\displaystyle X_{t})} are called hidden states , and P ⁡ ( Y n ∈ A ∣ X n = x n ) {\displaystyle \operatorname {\mathbf {P} } {\bigl (}Y_{n}\in A\mid X_{n}=x_{n}{\bigr )}} (resp. P ⁡ ( Y t ∈ A ∣ X t ∈ B t ) ) {\displaystyle \operatorname {\mathbf {P} } {\bigl (}Y_{t}\in A\mid X_{t}\in B_{t}{\bigr )})}

4235-472: Is a one-to-one function from R k {\displaystyle \mathbb {R} ^{k}} to itself, and reparameterize the likelihood function by setting ϕ i = h i ( θ 1 , θ 2 , … , θ k )   . {\displaystyle \;\phi _{i}=h_{i}(\theta _{1},\theta _{2},\ldots ,\theta _{k})~.} Because of

4356-424: Is a Markov chain for which the state is only partially observable or noisily observable. In other words, observations are related to the state of the system, but they are typically insufficient to precisely determine the state. Several well-known algorithms for hidden Markov models exist. For example, given a sequence of observations, the Viterbi algorithm will compute the most-likely corresponding sequence of states,

4477-432: Is a Markov decision process in which the state of the system is only partially observed. POMDPs are known to be NP complete , but recent approximation techniques have made them useful for a variety of applications, such as controlling simple agents or robots. A Markov random field , or Markov network, may be considered to be a generalization of a Markov chain in multiple dimensions. In a Markov chain, state depends only on

4598-637: Is a certain chance that Bob will perform one of the following activities, depending on the weather: "walk", "shop", or "clean". Since Bob tells Alice about his activities, those are the observations . The entire system is that of a hidden Markov model (HMM). Alice knows the general weather trends in the area, and what Bob likes to do on average. In other words, the parameters of the HMM are known. They can be represented as follows in Python : In this piece of code, start_probability represents Alice's belief about which state

4719-494: Is a column-vector of Lagrange multipliers and ∂ h ( θ ) T ∂ θ {\displaystyle \;{\frac {\partial h(\theta )^{\mathsf {T}}}{\partial \theta }}\;} is the k × r Jacobian matrix of partial derivatives. Naturally, if the constraints are not binding at the maximum, the Lagrange multipliers should be zero. This in turn allows for

4840-407: Is a method of estimating the parameters of an assumed probability distribution , given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model , the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood

4961-471: Is a model, often in idealized form, of the process generated by the data. It is a common aphorism in statistics that all models are wrong . Thus, true consistency does not occur in practical applications. Nevertheless, consistency is often considered to be a desirable property for an estimator to have. To establish consistency, the following conditions are sufficient. In other words, different parameter values θ correspond to different distributions within

Hidden Markov model - Misplaced Pages Continue

5082-625: Is a real upper triangular matrix and Γ T {\displaystyle \Gamma ^{\mathsf {T}}} is its transpose . In practice, restrictions are usually imposed using the method of Lagrange which, given the constraints as defined above, leads to the restricted likelihood equations where   λ = [ λ 1 , λ 2 , … , λ r ] T   {\displaystyle ~\lambda =\left[\lambda _{1},\lambda _{2},\ldots ,\lambda _{r}\right]^{\mathsf {T}}~}

5203-405: Is also of interest, one may alternatively resort to variational approximations to Bayesian inference, e.g. Indeed, approximate variational inference offers computational efficiency comparable to expectation-maximization, while yielding an accuracy profile only slightly inferior to exact MCMC-type Bayesian inference. HMMs can be applied in many fields where the goal is to recover a data sequence that

5324-412: Is both intuitive and flexible, and as such the method has become a dominant means of statistical inference . If the likelihood function is differentiable , the derivative test for finding maxima can be applied. In some cases, the first-order conditions of the likelihood function can be solved analytically; for instance, the ordinary least squares estimator for a linear regression model maximizes

5445-414: Is called emission probability or output probability . In its discrete form, a hidden Markov process can be visualized as a generalization of the urn problem with replacement (where each item from the urn is returned to the original urn before the next step). Consider this example: in a room that is not visible to an observer there is a genie. The room contains urns X1, X2, X3, ... each of which contains

5566-445: Is called the maximum likelihood estimator . It is generally a function defined over the sample space , i.e. taking a given sample as its argument. A sufficient but not necessary condition for its existence is for the likelihood function to be continuous over a parameter space Θ {\displaystyle \,\Theta \,} that is compact . For an open Θ {\displaystyle \,\Theta \,}

5687-464: 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

5808-413: 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" and was further formalized in "Hierarchical Dirichlet Processes". A different type of extension uses

5929-455: Is illustrated by the lower part of the diagram shown in Figure 1, where one can see that balls y1, y2, y3, y4 can be drawn at each state. Even if the observer knows the composition of the urns and has just observed a sequence of three balls, e.g. y1, y2 and y3 on the conveyor belt, the observer still cannot be sure which urn ( i.e. , at which state) the genie has drawn the third ball from. However,

6050-408: Is natural to ask about the state of the process at the end. This problem can be handled efficiently using the forward algorithm . An example is when the algorithm is applied to a Hidden Markov Network to determine P ( h t   | v 1 : t ) {\displaystyle \mathrm {P} {\big (}h_{t}\ |v_{1:t}{\big )}} . This

6171-444: Is not immediately observable (but other data that depend on the sequence are). Applications include: Hidden Markov models were described in a series of statistical papers by Leonard E. Baum and other authors in the second half of the 1960s. One of the first applications of HMMs was speech recognition , starting in the mid-1970s. From the linguistics point of view, hidden Markov models are equivalent to stochastic regular grammar. In

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6292-405: 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

6413-418: Is of interest is the entire sequence of parts of speech, rather than simply the part of speech for a single word, as filtering or smoothing would compute. This task requires finding a maximum over all possible state sequences, and can be solved efficiently by the Viterbi algorithm . For some of the above problems, it may also be interesting to ask about statistical significance . What is the probability that

6534-508: Is possible to estimate the second-order bias of the maximum likelihood estimator, and correct for that bias by subtracting it: This estimator is unbiased up to the terms of order ⁠ 1 /   n   ⁠ , and is called the bias-corrected maximum likelihood estimator . This bias-corrected estimator is second-order efficient (at least within the curved exponential family), meaning that it has minimal mean squared error among all second-order bias-corrected estimators, up to

6655-405: Is similar to filtering but asks about the distribution of a latent variable somewhere in the middle of a sequence, i.e. to compute P ( x ( k )   |   y ( 1 ) , … , y ( t ) ) {\displaystyle P(x(k)\ |\ y(1),\dots ,y(t))} for some k < t {\displaystyle k<t} . From

6776-498: Is taken with respect to the true density. Maximum-likelihood estimators have no optimum properties for finite samples, in the sense that (when evaluated on finite samples) other estimators may have greater concentration around the true parameter-value. However, like other estimation methods, maximum likelihood estimation possesses a number of attractive limiting properties : As the sample size increases to infinity, sequences of maximum likelihood estimators have these properties: Under

6897-440: Is that dynamic-programming algorithms for training them have an O ( N K T ) {\displaystyle O(N^{K}\,T)} running time, for K {\displaystyle K} adjacent states and T {\displaystyle T} total observations (i.e. a length- T {\displaystyle T} Markov chain). This extension has been widely used in bioinformatics , in

7018-403: Is that in finite samples, there may exist multiple roots for the likelihood equations. Whether the identified root θ ^ {\displaystyle \,{\widehat {\theta \,}}\,} of the likelihood equations is indeed a (local) maximum depends on whether the matrix of second-order partial and cross-partial derivatives, the so-called Hessian matrix

7139-565: Is the Fisher information matrix . The maximum likelihood estimator selects the parameter value which gives the observed data the largest possible probability (or probability density, in the continuous case). If the parameter consists of a number of components, then we define their separate maximum likelihood estimators, as the corresponding component of the MLE of the complete parameter. Consistent with this, if θ ^ {\displaystyle {\widehat {\theta \,}}}

7260-400: Is the MLE for θ {\displaystyle \theta } , and if g ( θ ) {\displaystyle g(\theta )} is any transformation of θ {\displaystyle \theta } , then the MLE for α = g ( θ ) {\displaystyle \alpha =g(\theta )} is by definition It maximizes

7381-542: Is the probability of the data averaged over all parameters. Since the denominator is independent of θ , the Bayesian estimator is obtained by maximizing f ( x 1 , x 2 , … , x n ∣ θ ) P ⁡ ( θ ) {\displaystyle f(x_{1},x_{2},\ldots ,x_{n}\mid \theta )\operatorname {\mathbb {P} } (\theta )} with respect to θ . If we further assume that

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7502-460: 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 settings, where the dependency structure enables identifiability of the model and

7623-401: The N × N {\displaystyle N\times N} matrix of transition probabilities is a Markov matrix . Because any transition probability can be determined once the others are known, there are a total of N ( N − 1 ) {\displaystyle N(N-1)} transition parameters. In addition, for each of the N possible states, there is

7744-453: The Cramér–Rao bound . Specifically, where   I   {\displaystyle ~{\mathcal {I}}~} is the Fisher information matrix : In particular, it means that the bias of the maximum likelihood estimator is equal to zero up to the order ⁠ 1 / √ n   ⁠ . However, when we consider the higher-order terms in the expansion of

7865-414: The Markov property . Similarly, the value of the observed variable y ( t ) depends on only the value of the hidden variable x ( t ) (both at time t ). In the standard type of hidden Markov model considered here, 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

7986-434: The covariance matrix , for a total of N ( M + M ( M + 1 ) 2 ) = N M ( M + 3 ) 2 = O ( N M 2 ) {\displaystyle N\left(M+{\frac {M(M+1)}{2}}\right)={\frac {NM(M+3)}{2}}=O(NM^{2})} emission parameters. (In such a case, unless the value of M is small, it may be more practical to restrict

8107-399: The forward algorithm will compute the probability of the sequence of observations, and the Baum–Welch algorithm will estimate the starting probabilities, the transition function, and the observation function of a hidden Markov model. One common use is for speech recognition , where the observed data is the speech audio waveform and the hidden state is the spoken text. In this example,

8228-534: The maximum a posteriori estimate is the parameter θ that maximizes the probability of θ given the data, given by Bayes' theorem: where P ⁡ ( θ ) {\displaystyle \operatorname {\mathbb {P} } (\theta )} is the prior distribution for the parameter θ and where P ⁡ ( x 1 , x 2 , … , x n ) {\displaystyle \operatorname {\mathbb {P} } (x_{1},x_{2},\ldots ,x_{n})}

8349-414: The Bayesian estimator coincides with the maximum likelihood estimator for a uniform prior distribution P ⁡ ( θ ) {\displaystyle \operatorname {\mathbb {P} } (\theta )} . In many practical applications in machine learning , maximum-likelihood estimation is used as the model for parameter estimation. The Bayesian Decision theory is about designing

8470-400: The HMM is in when Bob first calls her (all she knows is that it tends to be rainy on average). The particular probability distribution used here is not the equilibrium one, which is (given the transition probabilities) approximately {'Rainy': 0.57, 'Sunny': 0.43} . The transition_probability represents the change of the weather in the underlying Markov chain. In this example, there is only

8591-487: 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

8712-532: The HMM without the necessity of explicitly modeling the joint distribution, utilizing only the conditional distributions. 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

8833-448: The Viterbi algorithm finds the most likely sequence of spoken words given the speech audio. A Markov decision process is a Markov chain in which state transitions depend on the current state and an action vector that is applied to the system. Typically, a Markov decision process is used to compute a policy of actions that will maximize some utility with respect to expected rewards. A partially observable Markov decision process (POMDP)

8954-422: 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 K {\displaystyle K} adjacent states). The disadvantage of such models

9075-639: 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

9196-460: The conditions outlined below, the maximum likelihood estimator is consistent . The consistency means that if the data were generated by f ( ⋅ ; θ 0 ) {\displaystyle f(\cdot \,;\theta _{0})} and we have a sufficiently large number of observations n , then it is possible to find the value of θ 0 with arbitrary precision. In mathematical terms this means that as n goes to infinity

9317-411: The corresponding hidden variables of a set of K {\displaystyle K} independent Markov chains, rather than a single Markov chain. It is equivalent to a single HMM, with N K {\displaystyle N^{K}} states (assuming there are N {\displaystyle N} states for each chain), and therefore, learning in such a model is difficult: for

9438-430: The data are independent and identically distributed , then we have this being the sample analogue of the expected log-likelihood ℓ ( θ ) = E ⁡ [ ln ⁡ f ( x i ∣ θ ) ] {\displaystyle \ell (\theta )=\operatorname {\mathbb {E} } [\,\ln f(x_{i}\mid \theta )\,]} , where this expectation

9559-414: The data were generated by   f ( ⋅ ; θ 0 )   , {\displaystyle ~f(\cdot \,;\theta _{0})~,} then under certain conditions, it can also be shown that the maximum likelihood estimator converges in distribution to a normal distribution. It is √ n   -consistent and asymptotically efficient, meaning that it reaches

9680-403: The diagram (often called a trellis diagram ) denote conditional dependencies. From the diagram, it is clear that the conditional probability distribution of the hidden variable x ( t ) at time t , given the values of the hidden variable x at all times, depends only on the value of the hidden variable x ( t − 1) ; the values at time t − 2 and before have no influence. This is called

9801-404: 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. Yet another variant is the factorial hidden Markov model , which allows for a single observation to be conditioned on

9922-420: The distinction between B 1 , B 2 {\displaystyle B_{1},B_{2}} , this space of subshifts is projected on A , B 1 , B 2 {\displaystyle A,B_{1},B_{2}} into another space of subshifts on A , B {\displaystyle A,B} , and this projection also projects the probability measure down to

10043-464: The distribution of this estimator, it turns out that θ mle has bias of order 1 ⁄ n . This bias is equal to (componentwise) where I j k {\displaystyle {\mathcal {I}}^{jk}} (with superscripts) denotes the ( j,k )-th component of the inverse Fisher information matrix I − 1 {\displaystyle {\mathcal {I}}^{-1}} , and Using these formulae it

10164-549: The equivariance of the maximum likelihood estimator, the properties of the MLE apply to the restricted estimates also. For instance, in a multivariate normal distribution the covariance matrix Σ {\displaystyle \,\Sigma \,} must be positive-definite ; this restriction can be imposed by replacing Σ = Γ T Γ , {\displaystyle \;\Sigma =\Gamma ^{\mathsf {T}}\Gamma \;,} where Γ {\displaystyle \Gamma }

10285-400: The estimation process. The parameter space can be expressed as where h ( θ ) = [ h 1 ( θ ) , h 2 ( θ ) , … , h r ( θ ) ] {\displaystyle \;h(\theta )=\left[h_{1}(\theta ),h_{2}(\theta ),\ldots ,h_{r}(\theta )\right]\;} is

10406-565: The estimator θ ^ {\displaystyle {\widehat {\theta \,}}} converges in probability to its true value: Under slightly stronger conditions, the estimator converges almost surely (or strongly ): In practical applications, data is never generated by f ( ⋅ ; θ 0 ) {\displaystyle f(\cdot \,;\theta _{0})} . Rather, f ( ⋅ ; θ 0 ) {\displaystyle f(\cdot \,;\theta _{0})}

10527-422: The general architecture of an instantiated HMM. Each oval shape represents a random variable that can adopt any of a number of values. The random variable x ( t ) is the hidden state at time t (with the model from the above diagram, x ( t ) ∈ { x 1 , x 2 , x 3 }) . The random variable y ( t ) is the observation at time t (with y ( t ) ∈ { y 1 , y 2 , y 3 , y 4 }) . The arrows in

10648-402: The goal is to learn about state of X {\displaystyle X} by observing Y {\displaystyle Y} . By definition of being a Markov model, an HMM has an additional requirement that the outcome of Y {\displaystyle Y} at time t = t 0 {\displaystyle t=t_{0}} must be "influenced" exclusively by

10769-401: The hypothesis for the output sequence. 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

10890-508: The joint density at the observed data sample y = ( y 1 , y 2 , … , y n ) {\displaystyle \;\mathbf {y} =(y_{1},y_{2},\ldots ,y_{n})\;} gives a real-valued function, which is called the likelihood function . For independent and identically distributed random variables , f n ( y ; θ ) {\displaystyle f_{n}(\mathbf {y} ;\theta )} will be

11011-674: The latent Markov models, with special attention to the model assumptions and to their practical use is provided in Given a Markov transition matrix and an invariant distribution on the states, a probability measure can be imposed on the set of subshifts. For example, consider the Markov chain given on the left on the states A , B 1 , B 2 {\displaystyle A,B_{1},B_{2}} , with invariant distribution π = ( 2 / 7 , 4 / 7 , 1 / 7 ) {\displaystyle \pi =(2/7,4/7,1/7)} . By ignoring

11132-408: The learnability limits are still under exploration. 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

11253-419: The likelihood function L n {\displaystyle \,{\mathcal {L}}_{n}\,} is called the maximum likelihood estimate. Further, if the function θ ^ n : R n → Θ {\displaystyle \;{\hat {\theta }}_{n}:\mathbb {R} ^{n}\to \Theta \;} so defined is measurable , then it

11374-411: The likelihood function may increase without ever reaching a supremum value. In practice, it is often convenient to work with the natural logarithm of the likelihood function, called the log-likelihood : Since the logarithm is a monotonic function , the maximum of ℓ ( θ ; y ) {\displaystyle \;\ell (\theta \,;\mathbf {y} )\;} occurs at

11495-409: The likelihood when the random errors are assumed to have normal distributions with the same variance. From the perspective of Bayesian inference , MLE is generally equivalent to maximum a posteriori (MAP) estimation with a prior distribution that is uniform in the region of interest. In frequentist inference , MLE is a special case of an extremum estimator , with the objective function being

11616-714: The likelihood. We model a set of observations as a random sample from an unknown joint probability distribution which is expressed in terms of a set of parameters . The goal of maximum likelihood estimation is to determine the parameters for which the observed data have the highest joint probability. We write the parameters governing the joint distribution as a vector θ = [ θ 1 , θ 2 , … , θ k ] T {\displaystyle \;\theta =\left[\theta _{1},\,\theta _{2},\,\ldots ,\,\theta _{k}\right]^{\mathsf {T}}\;} so that this distribution falls within

11737-498: The maximum of the likelihood function subject to the constraint   h ( θ ) = 0   . {\displaystyle ~h(\theta )=0~.} Theoretically, the most natural approach to this constrained optimization problem is the method of substitution, that is "filling out" the restrictions h 1 , h 2 , … , h r {\displaystyle \;h_{1},h_{2},\ldots ,h_{r}\;} to

11858-570: The model allow for faster learning and inference. A Tolerant Markov model (TMM) is a probabilistic-algorithmic Markov chain model. It assigns the probabilities according to a conditioning context that considers the last symbol, from the sequence to occur, as the most probable instead of the true occurring symbol. A TMM can model three different natures: substitutions, additions or deletions. Successful applications have been efficiently implemented in DNA sequences compression. Markov-chains have been used as

11979-412: The model. If this condition did not hold, there would be some value θ 1 such that θ 0 and θ 1 generate an identical distribution of the observable data. Then we would not be able to distinguish between these two parameters even with an infinite amount of data—these parameters would have been observationally equivalent . The identification condition establishes that the log-likelihood has

12100-605: The modeling of DNA sequences . Another recent extension is the triplet Markov model , 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 and which allows to fuse data in Markovian context and to model nonstationary data. Alternative multi-stream data fusion strategies have also been proposed in recent literature, e.g., Finally,

12221-405: The nature of the covariances between individual elements of the observation vector, e.g. by assuming that the elements are independent of each other, or less restrictively, are independent of all but a fixed number of adjacent elements.) Several inference problems are associated with hidden Markov models, as outlined below. The task is to compute in a best way, given the parameters of the model,

12342-544: The non-i.i.d. case, the uniform convergence in probability can be checked by showing that the sequence ℓ ^ ( θ ∣ x ) {\displaystyle {\widehat {\ell \,}}(\theta \mid x)} is stochastically equicontinuous . If one wants to demonstrate that the ML estimator θ ^ {\displaystyle {\widehat {\theta \,}}} converges to θ 0 almost surely , then

12463-486: The observation's law. 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. 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

12584-423: The observer can work out other information, such as the likelihood that the third ball came from each of the urns. Consider two friends, Alice and Bob, who live far apart from each other and who talk together daily over the telephone about what they did that day. Bob is only interested in three activities: walking in the park, shopping, and cleaning his apartment. The choice of what to do is determined exclusively by

12705-430: The occurrence of a maximum (or a minimum) are known as the likelihood equations. For some models, these equations can be explicitly solved for θ ^ , {\displaystyle \,{\widehat {\theta \,}}\,,} but in general no closed-form solution to the maximization problem is known or available, and an MLE can only be found via numerical optimization . Another problem

12826-654: The outcome of X {\displaystyle X} at t = t 0 {\displaystyle t=t_{0}} and that the outcomes of X {\displaystyle X} and Y {\displaystyle Y} at t < t 0 {\displaystyle t<t_{0}} must be conditionally independent of Y {\displaystyle Y} at t = t 0 {\displaystyle t=t_{0}} given X {\displaystyle X} at time t = t 0 {\displaystyle t=t_{0}} . Estimation of

12947-441: 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

13068-887: The parameters in an HMM can be performed using maximum likelihood estimation . For linear chain HMMs, the Baum–Welch algorithm can be used to estimate parameters. Hidden Markov models are known for their applications to thermodynamics , statistical mechanics , physics , chemistry , economics , finance , signal processing , information theory , pattern recognition —such as speech , handwriting , gesture recognition , part-of-speech tagging , musical score following, partial discharges and bioinformatics . Let X n {\displaystyle X_{n}} and Y n {\displaystyle Y_{n}} be discrete-time stochastic processes and n ≥ 1 {\displaystyle n\geq 1} . The pair ( X n , Y n ) {\displaystyle (X_{n},Y_{n})}

13189-412: The perspective described above, this can be thought of as the probability distribution over hidden states for a point in time k in the past, relative to time t . The forward-backward algorithm is a good method for computing the smoothed values for all hidden state variables. The task, unlike the previous two, asks about the joint probability of the entire sequence of hidden states that generated

13310-419: The previous state in time, whereas in a Markov random field, each state depends on its neighbors in any of multiple directions. A Markov random field may be visualized as a field or graph of random variables, where the distribution of each random variable depends on the neighboring variables with which it is connected. More specifically, the joint distribution for any random variable in the graph can be computed as

13431-445: The prior P ⁡ ( θ ) {\displaystyle \operatorname {\mathbb {P} } (\theta )} is a uniform distribution, the Bayesian estimator is obtained by maximizing the likelihood function f ( x 1 , x 2 , … , x n ∣ θ ) {\displaystyle f(x_{1},x_{2},\ldots ,x_{n}\mid \theta )} . Thus

13552-404: The probability of a particular output sequence. This requires summation over all possible state sequences: The probability of observing a sequence of length L is given by where the sum runs over all possible hidden-node sequences Applying the principle of dynamic programming , this problem, too, can be handled efficiently using the forward algorithm . A number of related tasks ask about

13673-400: The probability of one or more of the latent variables, given the model's parameters and a sequence of observations y ( 1 ) , … , y ( t ) {\displaystyle y(1),\dots ,y(t)} . The task is to compute, given the model's parameters and a sequence of observations, the distribution over hidden states of the last latent variable at the end of

13794-429: The product of the "clique potentials" of all the cliques in the graph that contain that random variable. Modeling a problem as a Markov random field is useful because it implies that the joint distributions at each vertex in the graph may be computed in this manner. Hierarchical Markov models can be applied to categorize human behavior at various levels of abstraction. For example, a series of simple observations, such as

13915-598: The product of univariate density functions : The goal of maximum likelihood estimation is to find the values of the model parameters that maximize the likelihood function over the parameter space, that is Intuitively, this selects the parameter values that make the observed data most probable. The specific value   θ ^ = θ ^ n ( y ) ∈ Θ   {\displaystyle ~{\hat {\theta }}={\hat {\theta }}_{n}(\mathbf {y} )\in \Theta ~} that maximizes

14036-455: The same value of θ {\displaystyle \theta } as does the maximum of L n   . {\displaystyle \,{\mathcal {L}}_{n}~.} If ℓ ( θ ; y ) {\displaystyle \ell (\theta \,;\mathbf {y} )} is differentiable in Θ , {\displaystyle \,\Theta \,,} sufficient conditions for

14157-434: The second half of the 1980s, HMMs began to be applied to the analysis of biological sequences, in particular DNA . Since then, they have become ubiquitous in the field of bioinformatics . 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

14278-438: The sequence, i.e. to compute P ( x ( t )   |   y ( 1 ) , … , y ( t ) ) {\displaystyle P(x(t)\ |\ y(1),\dots ,y(t))} . This task is used when the sequence of latent variables is thought of as the underlying states that a process moves through at a sequence of points in time, with corresponding observations at each point. Then, it

14399-506: The so-called profile likelihood : The MLE is also equivariant with respect to certain transformations of the data. If y = g ( x ) {\displaystyle y=g(x)} where g {\displaystyle g} is one to one and does not depend on the parameters to be estimated, then the density functions satisfy and hence the likelihood functions for X {\displaystyle X} and Y {\displaystyle Y} differ only by

14520-415: The terms of the order ⁠ 1 /   n   ⁠  . It is possible to continue this process, that is to derive the third-order bias-correction term, and so on. However, the maximum likelihood estimator is not third-order efficient. A maximum likelihood estimator coincides with the most probable Bayesian estimator given a uniform prior distribution on the parameters . Indeed,

14641-424: The weather on a given day. Alice has no definite information about the weather, but she knows general trends. Based on what Bob tells her he did each day, Alice tries to guess what the weather must have been like. Alice believes that the weather operates as a discrete Markov chain . There are two states, "Rainy" and "Sunny", but she cannot observe them directly, that is, they are hidden from her. On each day, there

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