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Sequential Modeling with the Hidden Markov Model

Sequential Modeling with the Hidden Markov Model. Lecture 9 Spoken Language Processing Prof. Andrew Rosenberg. Markov Assumption. If we can represent all of the information available in the present state, encoding the past is un-necessary.

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Sequential Modeling with the Hidden Markov Model

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  1. Sequential Modeling with the Hidden Markov Model Lecture 9 Spoken Language Processing Prof. Andrew Rosenberg

  2. Markov Assumption • If we can represent all of the information available in the present state, encoding the past is un-necessary. The future is independent of the past given the present

  3. Markov Assumption in Speech • Word Sequences • Phone Sequences • Part of Speech Tags • Syntactic constituents • Phrase sequences • Discourse Acts • Intonation

  4. Markov Chain • The probability of a sequence can be decomposed into a probability of sequential events. x1 x2 x3

  5. Hidden Markov model • In a Hidden Markov Model the state sequence is unobserved. • Only an observation sequence is available q1 q2 q3 x1 x2 x3

  6. Hidden Markov model • Observations are MFCC vectors • States are phone labels • Each state (phone) has an associated GMM modeling the MFCC likelihood q1 q2 q3 x1 x2 x3

  7. Forward-Backwards Algorithm • HMMs are trained by collecting and distributing information from observations to states. • The Forward-Backwards algorithm is a specific example of EM. • In the HMM topology (variable relationship), the training converges in one forward pass, and a backwards pass. • hence the name

  8. Forwards Backwards Algorithm • Forwards-Step: • Collect up from the observations to the states • Collect from left state to right state. • “Collect” – update parameters to correctly model the observations • Observation collection will give a distribution over states, given the initial state • State collection will also give a distribution over states • the new q distribution will reflect the combination of these two q1 q2 q3 x1 x2 x3

  9. Forwards Backwards Algorithm • Backwards-Step: • Distribute down to the observations from the states • Collect from left state to right state. • “Distribute” – update parameters to correctly model the observations • Observation distribute updates the state-observation relationship • State distribution updates the state-state transition matrix • Forward-backwards can be shown to converge in one pass. q1 q2 q3 x1 x2 x3

  10. Finite State Automata • “Start” “Accept” States • Epsilon Transitions • Relationship to Regular Expressions • Operations on FSA • Addition • Inversion • Node expansion • Determinization • Weighted automata allow probabilities to be assigned to transitions

  11. State transitions as FSA /d/ /t/ /ey/ /ax/ /ae/ /dx/

  12. Word FSA to phone FSA MORE DATA /d/ /t/ /ey/ /ax/ /ae/ /dx/ /m/ /ao/ /r/

  13. Word FSA to phone FSA /d/ /t/ /ey/ /ax/ /ae/ /dx/ /m/ /ao/ /r/

  14. Decoding a Hidden Markov Model • Decoding is finding the most likely state sequence. • How many state sequences are there in a HMM with N observations and k states?

  15. Viterbi Decoding • Dynamic Programming can make this a lot faster. • Idea: Any optimal sequence between x0 and xnmust include the optimal sequence between xn and xn-1. • Based on the Markov Assumption.

  16. Viterbi Decoding • Probability of most likely state sequence • Recovering the the optimal sequence involves storing pointers as decisions are made.

  17. Example (from Wikipedia) states = ('Rainy', 'Sunny') observations = ('walk', 'shop', 'clean') start_probability = {'Rainy': 0.6, 'Sunny': 0.4} transition_probability = { 'Rainy' : {'Rainy': 0.7, 'Sunny': 0.3}, 'Sunny' : {'Rainy': 0.4, 'Sunny': 0.6}, } emission_probability = { 'Rainy' : {'walk': 0.1, 'shop': 0.4, 'clean': 0.5}, 'Sunny' : {'walk': 0.6, 'shop': 0.3, 'clean': 0.1}, } What is the most likely state sequence?

  18. HMM Topology for Training • Rather than having one GMM per phone, it is common for acoustic models to represent each phone as 3 triphones S3 S2 S4 /r/ S5 S1

  19. Flat Start • In Flat Start training, GMM parameters are initialized to global means and variances. • Viterbi is used to perform forced alignmentbetween observations and phone sequence. • The phone sequence is derived from the lexical transcription and pronunciation model

  20. Forced Alignment • Given a phone sequence and observations, assign each observation to a phone. • Uses • Identifying which observation belong to each phone label for later training • Getting time boundaries for phone or word labels.

  21. Flat Start • In Flat Start training, GMM parameters are initialized to global means and variances. • Viterbi is used to perform forced alignmentbetween observations and phone sequence. • The phone sequence is derived from the lexical transcription and pronunciation model • After alignment, retrain Acoustic Models, and repeat.

  22. What about silence? • If there is no “silence” state, the silent frames will be assigned to either the /d/ or the /ax/ • This leads to worse acoustic models. • A solution: Explicit training of silence models, /sp/ • Allowing /sp/ transitions at word boundaries /d/ /ey/ /dx/ /ax/ /d/ /ey/ /dx/ /ax/ /d/ /ey/ /dx/ /ax/

  23. Next Class • Pronunciation Modeling • Reading: J&M Chapter 2, Section10.5.3, 11.1, 11.2

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