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Hidden Markov Model

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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)

- limited horizon:
- 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