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# Hidden Markov Model - PowerPoint PPT Presentation

Hidden Markov Model. Three Holy Questions of Markov. EHSAN KHODDAM MOHAMMADI. Markov Model (1/). Used when modeling a sequence of random variables that aren’t independent the value of each variable depends only on the previous element in the sequence

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### Hidden Markov Model

Three Holy Questions of Markov

EHSAN KHODDAM MOHAMMADI

• Used when modeling a sequence of random variables that

• aren’t independent

• the value of each variable depends only on the previous element in the sequence

• In other words, in a Markov model, future elements are conditionally independent of the past elements given the present element

• Actually HMM is an Automata

• States

• Probabailstic rules for state-transition

• State-Emission

• Formal definition will come later

• In an HMM, you don’t know the state sequence that the model passes through, but only some probabilistic function of it

• You Model the underlying Dynamics of a process which generating surface events, You don’t know what’s going on but you could predict it!

• X = (X1, …, XT): a sequence of random variables,

Taking values in some finite set S = {s1, …, sN}

or the state space

• Markov properties

• limited horizon:

• P(Xt+1 = sk| X1,t) = P(Xt+1 = sk| Xt)

• time invariant (stationary):

• P(Xt+1 = Si| Xt = sj) = P(X2 = si| X1 = sj)

• X: a Markov chain

• Transition matrix, A = [ aij ]

aij = P(Xt+1 = si| Xt = sj)

• Initial state probabilities, Π = [ πi ]

πi = P(X1 = si)

• Probability of a Markov chain X = (X1, …, XT)

P(X1, …, XT) = P(X1) P(X2|X1) … P(XT|XT-1)

πX1aX2.X1 … aXTXT-1

= πX1Πt=1,T-1aXtXt-1

0.3

0.7

0.5

0.5

• 1. Given a model = (A, B,П), how do we efficiently compute how likely a certain observation is, that is P(O|μ)?

• 2. Given the observation sequence O and a model μ how do we choose a state sequence X1, …, XT, that best explains the observations?

• 3. Given an observation sequence O, and a space of possible models found by varying the model parameters = (A, B, П), how do we find the model that best explains the observed data?

• What is the probability of seeing the output sequence {lem, ice_t} if the machine always starts off in the cola preferring state?

• Forward Algorithm(D.P.)

• Viterbi Algorithm (D.P.)

• Baum-Welch Algorithm (EM optimization)

• “Foundations Of Statistical Natural Language Processing”, Ch 9, Manning & Schutze , 2000

• “Hidden Markov Models for Time Series - An Introduction Using R”, Zucchini, 2009

• “NLP 88 Class lectures” , CSE, Shiraz University, Dr. Fazli, 2009