Pattern Classification

All materials in these slides were taken from

Pattern Classification (2nd ed) by R. O. Duda, P. E. Hart and D. G. Stork, John

Wiley & Sons, 2000 with the permission of the authors and

the publisher

Chapter 3 (Part 3): Maximum-Likelihood and Bayesian

Parameter Estimation (Section 3.10)

• Hidden Markov Model: Extension of

Markov Chains

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Pattern Classification, Chapter 3 (Part 3)

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• Hidden Markov Model (HMM)

• Interaction of the visible states with the hidden states

bjk= 1 for all j where bjk=P(Vk(t) | j(t)).

• 3 problems are associated with this model

• The evaluation problem

• The decoding problem

• The learning problem

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•The evaluation problem

It is the probability that the model produces a sequence VT of visible states. It is:

where each r indexes a particular sequence

of T hidden states.

)(P)|V(P)V(P Tr

r

1r

Tr

TTmax

)T(),...,2(),1(Tr

Tt

1t

Tr

Tt

1t

Tr

T

))1t(|)t((P)P( (2)

))t(|)t(v(P)|V(P )1(

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Using equations (1) and (2), we can write:

Interpretation: The probability that we observe the particular sequence of T visible states VT is equal to the sum over all rmax possible sequences of hidden states of the conditional probability that the system has made a particular transition multiplied by the probability that it then emitted the visible symbol in our target sequence.

Example: Let 1, 2, 3 be the hidden states; v1, v2, v3 be the visible states

and V3 = {v1, v2, v3} is the sequence of visible states

P({v1, v2, v3}) = P(1).P(v1 | 1).P(2 | 1).P(v2 | 2).P(3 | 2).P(v3 | 3)+…+ (possible terms in the sum= all possible (33= 27) cases

!)

maxr

1r

Tt

1t

T )1t(|)t((P ))t(|)t(v(P)V(P

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First possibility:

Second Possibility:

P({v1, v2, v3}) = P(2).P(v1 | 2).P(3 | 2).P(v2 | 3).P(1 | 3).P(v3 | 1) + …+

Therefore:

states hidden ofsequence possible

3t

1t321 ))1t(|)t((P)).t(|)t(v(P})v,v,v({P

1

(t = 1)2

(t = 2)

3

(t = 3)

v1 v2 v3

2

(t = 1)1

(t = 3)3

(t = 2)

v3v2v1

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Evaluation

• HMM forward

• HMM backward

• Example 3.

• HMM Forward

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Pattern Classification, Chapter 3 (Part 3)

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Left-to-Right model (speech recognition)

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•The decoding problem (optimal state sequence)

Given a sequence of visible states VT, the decoding problem is to find the most probable sequence of hidden states.

This problem can be expressed mathematically as:

find the single “best” state sequence (hidden states)

|)T(V),...,2(v),1(v),T(),...,2(),1(Pmaxarg)T(ˆ),...,2(ˆ),1(ˆ

:that such )T(ˆ),...,2(ˆ),1(ˆ

)T(),...,2(),1(

Note that the summation disappeared, since we want to findOnly one unique best case !

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Where: = [,A,B]

= P((1) = ) (initial state probability)

A = aij = P((t+1) = j | (t) = i)

B = bjk = P(v(t) = k | (t) = j)

In the preceding example, this computation corresponds to the selection of the best path amongst:

{1(t = 1),2(t = 2),3(t = 3)}, {2(t = 1),3(t = 2),1(t = 3)}

{3(t = 1),1(t = 2),2(t = 3)}, {3(t = 1),2(t = 2),1(t = 3)}

{2(t = 1),1(t = 2),3(t = 3)}

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Decoding

• HMM decoding

• Example 4

• Might have invalid path

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•The learning problem (parameter estimation)

This third problem consists of determining a method to adjust the model parameters = [,A,B] to satisfy a certain optimization criterion. We need to find the best model

Such that to maximize the probability of the observation sequence:

We use an iterative procedure such as Baum-Welch or Gradient to find this local optimum

]ˆ,ˆ,ˆ[^

BA

)|(

TVPMax

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Learning

• The Forward-Backward Algorithm

• Baum-Welch Algorithm

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bby Eq. 140 y Eq. 140

bby Eq. 141y Eq. 141