Hidden Markov Models By Manish Shrivastava.

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Presentation transcript:

Hidden Markov Models By Manish Shrivastava

Hidden Markov Model U1 U2 U3 0.1 0.4 0.5 0.6 0.2 0.3 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: U1 U2 U3 0.1 0.4 0.5 0.6 0.2 0.3

Hidden Markov Model Set of states : S where |S|=N Output Alphabet : V Transition Probabilities : A = {aij} Emission Probabilities : B = {bj(ok)} Initial State Probabilities : π

Three Basic Problems of HMM Given Observation Sequence O ={o1… oT} Efficiently estimate P(O|λ) Get best Q ={q1… qT} i.e. Maximize P(Q|O, λ) How to adjust to best maximize Re-estimate λ

Solutions Problem 1: Likelihood of a sequence Forward Procedure Backward Procedure Problem 2: Best state sequence Viterbi Algorithm Problem 3: Re-estimation Baum-Welch ( Forward-Backward Algorithm )

Problem 1

Problem 1 Order 2TNT Definitely not efficient!! Is there a method to tackle this problem? Yes. Forward or Backward Procedure

Forward Procedure Forward Step:

Forward Procedure

Backward Procedure

Backward Procedure

Forward Backward Procedure Benefit Order N2T as compared to 2TNT for simple computation Only Forward or Backward procedure needed for Problem 1

Problem 2 Given Observation Sequence O ={o1… oT} Solution : Get “best” Q ={q1… qT} i.e. Solution : Best state individually likely at a position i Best state given all the previously observed states and observations Viterbi Algorithm

Viterbi Algorithm Define such that, i.e. the sequence which has the best joint probability so far. By induction, we have,

Viterbi Algorithm

Viterbi Algorithm

Problem 3 How to adjust to best maximize Solutions : Re-estimate λ To re-estimate (iteratively update and improve) HMM parameters A,B, π Use Baum-Welch algorithm

Baum-Welch Algorithm Define Putting forward and backward variables

Baum-Welch algorithm

Define Then, expected number of transitions from Si And, expected number of transitions from Sj to Si

Baum-Welch Algorithm Baum et al have proved that the above equations lead to a model as good or better than the previous

Questions?