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3.5 Markov Models 61yy yt-1 tt+1xt-1xtxt+1Fig. 3.19. A hidden Markov model with observations {x n } and latent states {y n }3.5.2 Hidden Markov ModelsA hidden Markov model (HMM) is a statistical model in which the system being modeledis assumed to be a Markov process with unobserved state [2]. In a regular Markov model (asintroduced in the last section), the state can be visible to the observer directly and thus, theonly parameter in the model is the state transition probability. However, in a hidden Markovmodel, the state can not be directly visible (i.e., hidden), but the event (or observation, output)which is dependent on the state, is visible. In other words, the state behaves like a latentvariable, where the latent variable is discrete. The hidden Markov model can be illustrated ina graph, as shown in Figure 3.19.The intuitive idea of HMM is that, because every state has a probability distribution overthe possible output, the output sequence generated by an HMM provides some related informationabout the state sequence.Hidden Markov models are especially known for their application in temporal patternrecognition such as speech [125], handwriting [187], natural language processing [179], musicalscore following, partial discharges and bioinformatics [141].Suppose that the hidden state behaves like a discrete multinomial variable y n and it controlshow to generate the corresponding event x n . The probability distribution of y n is dependenton the previous latent variable y n−1 and thus, we have a conditional distributionp(y n |y n−1 ). We assume that the latent variables having S states so that the conditional distributioncorresponds to a set of numbers which named transition probabilities. All of thesenumbers together are denoted by T and can be represented as T jk ≡ p(y nk = 1|y n−1, j = 1),where we have 0 ≤ T jk ≤ 1 with ∑ k T jk = 1. Therefore, the matrix T has S(S −1) independentparameters. The conditional distribution can be computed asS Sp(y n |y n−1,T )= ∏ ∏ T y n−1, jy nkjk . (3.8)k=1 j=1Note that for the initial latent variable y 1 , because it does not have a previous variable, theabove equation can be adapted as p(y 1 |π)=∏ S k=1 πy 1kk , where π denotes a vector of probabilitieswith π k ≡ p(y 1k = 1) and ∑ k π k = 1.Emission probabilities are defined as the conditional distributions of the output p(x n |y n ,φ),where φ is a set of parameters controlling the distribution. These probabilities could be modeledby conditional probabilities if x is discrete, or by Gaussians if the elements of x arecontinuous variables. The emission probabilities can be represented asSp(x n |y n ,φ)= ∏ p(x n |φ k ) y nk. (3.9)k=1