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.

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Presentation transcript:

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

Hidden Markov Model: Extension of Markov Chains Chapter 3 (Part 3): Maximum-Likelihood and Bayesian Parameter Estimation (Section 3.10) Hidden Markov Model: Extension of Markov Chains

bjk= 1 for all j where bjk=P(Vk(t) | j(t)). 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 Pattern Classification, Chapter 3 (Part 3)

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. Pattern Classification, Chapter 3 (Part 3)

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 !) Pattern Classification, Chapter 3 (Part 3)

First possibility: Second Possibility: Therefore: v1 v2 v3 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: 1 (t = 1) 2 (t = 2) 3 (t = 3) v1 v2 v3 2 (t = 1) 3 (t = 2) 1 (t = 3) Pattern Classification, Chapter 3 (Part 3)

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) Note that the summation disappeared, since we want to find Only one unique best case ! Pattern Classification, Chapter 3 (Part 3)

 = P((1) = ) (initial state probability) 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)} Pattern Classification, Chapter 3 (Part 3)

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 Pattern Classification, Chapter 3 (Part 3)