Introduction to HMM (cont)

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Presentation transcript:

Introduction to HMM (cont) CHEN TZAN HWEI Reference : the slides of prof. Berlin Chen

Forward procedure state s1 s1 s1 … s2 s2 s2 s3 s3 s3 time o1 o2 … oT 2019/4/30 Speech Lab. NTNU

backward procedure state s1 s1 s1 … s2 s2 s2 s3 s3 s3 time o1 … oT-1 2019/4/30 Speech Lab. NTNU

Basis problem 2 of problem How to choose the optimal state sequence? Why? Assuming that a state is a word state 天氣天氣天氣 … 氣象氣象氣象天象天象天象 time o1 … oT-1 oT 2019/4/30 Speech Lab. NTNU

Basis problem 2 of problem (cont) The intuitive criterion : choose the state i are individually most likely at each time t The question : invalid state sequence , ex: 2019/4/30 Speech Lab. NTNU

Basis problem 2 of problem (cont) Solution : Viterbi algorithm, can be consider as a modify forward algorithm 2019/4/30 Speech Lab. NTNU

Basis problem 2 of problem (cont) Algorithm : Define a new variable : Induction step : We can backtrace from 2019/4/30 Speech Lab. NTNU

Basis problem 2 of problem (cont) Algorithm in logarithmic domain 2019/4/30 Speech Lab. NTNU

Probability addition in F-B algorithm Assume we want to add and 2019/4/30 Speech Lab. NTNU

Probability addition in F-B algorithm 2019/4/30 Speech Lab. NTNU

Basis problem 3 of problem How to adjust (re-estimate) the model parameter to maximize The most difficult of the three problem : there no known analytical method that maximize the joint probability of the training data in a close form. The data is incomplete because of the hidden state sequences Well-solved by Baum-Welch (known as forward-backward) algorithm and EM (Expectation-Maximization) algorithm. Iterative update and improvement 2019/4/30 Speech Lab. NTNU

Basis problem 3 of problem (cont) 2019/4/30 Speech Lab. NTNU

Basis problem 3 of problem (cont) Intuitive view : 2019/4/30 Speech Lab. NTNU

Basis problem 3 of problem (cont) How to calculate the expected probability in state i at time t Define a new variable : Expected probability in state i at time “1” -> 2019/4/30 Speech Lab. NTNU

Basis problem 3 of problem (cont) How to calculate the expected probability of transition from state i to state j? Define a new variable : expected probability of transition from state i to state j -> 2019/4/30 Speech Lab. NTNU

Basis problem 3 of problem (cont) How to calculate the expected probability of transition from state i ? We observe that all transition from state i -> So, the meaning of “all transition from state i” is the same as that of “in state i”. 2019/4/30 Speech Lab. NTNU

Basis problem 3 of problem (cont) How to calculate the expected probability in state i and observing symbol 2019/4/30 Speech Lab. NTNU

Basis problem 3 of problem (cont) Summary For single training utterance 2019/4/30 Speech Lab. NTNU

Basis problem 3 of problem (cont) Summary For multiple (L) training utterances 2019/4/30 Speech Lab. NTNU