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Hidden Markov Models PowerPoint PPT Presentation

Hidden Markov Models. By Manish Shrivastava. A Simple Process. b. S1. b. a. S2. a. A simple automata. A Slightly Complicated Process. A colored ball choosing example :. Urn 1 # of Red = 100 # of Green = 0 # of Blue = 0. Urn 3 # of Red = 0 # of Green = 0 # of Blue = 100.

Hidden Markov Models

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Hidden Markov Models

By

Manish Shrivastava

A Simple Process

b

S1

b

a

S2

a

A simple automata

A Slightly Complicated Process

A colored ball choosing example :

Urn 1

# of Red = 100

# of Green = 0

# of Blue = 0

Urn 3

# of Red = 0

# of Green = 0

# of Blue = 100

Urn 2

# of Red = 0

# of Green = 100

# of Blue = 0

Probability of transition to another Urn after picking a ball:

A Slightly Complicated Process contd.

Given :

Observation : RRGGBRGR

State Sequence : ??

Easily Computable.

Markov Processes

• Properties

• Limited Horizon :Given previous n states, a state i, is independent of preceding 0…i-n+1 states.

• P(Xt=i|Xt-1, Xt-2 ,…X0) = P(Xt=i|Xt-1, Xt-2… Xt-n)

• Time invariance :

• P(Xt=i|Xt-1=j) = P(X1=i|X0=j) = P(Xn=i|X0-1=j)

A (Slightly Complicated) Markov Process

A colored ball choosing example :

Urn 1

# of Red = 100

# of Green = 0

# of Blue = 0

Urn 3

# of Red = 0

# of Green = 0

# of Blue = 100

Urn 2

# of Red = 0

# of Green = 100

# of Blue = 0

Probability of transition to another Urn after picking a ball:

Markov Process

• Visible Markov Model

• Given the observation, one can easily follow the state sequence traversed

1

2

3

P(1|3)

P(3|1)

Hidden Markov Model

A colored ball choosing example :

Urn 1

# of Red = 30

# of Green = 50

# of Blue = 20

Urn 3

# of Red =60

# of Green =10

# of Blue = 30

Urn 2

# of Red = 10

# of Green = 40

# of Blue = 50

Probability of transition to another Urn after picking a ball:

Hidden Markov Model

Given :

and

Observation : RRGGBRGR

State Sequence : ??

Not so Easily Computable.

Hidden Markov Model

• Set of states : S

• Output Alphabet : V

• Transition Probabilities : A = {aij}

• Emission Probabilities : B = {bj(ok)}

• Initial State Probabilities : π

Hidden Markov Model

• Here :

• S = {U1, U2, U3}

• V = { R,G,B}

• For observation:

• O ={o1… on}

• And State sequence

• Q ={q1… qn}

• π is

A =

B=

Three Basic Problems of HMM

• Given Observation Sequence O ={o1… on}

• Efficiently estimate

• Given Observation Sequence O ={o1… on}

• Get best Q ={q1… qn}

• How to adjust to best maximize

Questions?

References

• Lawrence R. Rabiner, A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition. Proceedings of the IEEE, 77 (2), p. 257–286, February 1989.

• Chris Manning and Hinrich Schütze, Chapter 9: Markov Models,Foundations of Statistical Natural Language Processing, MIT Press. Cambridge, MA: May 1999