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

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Markov Assumption If we can represent all of the information available in the present state, encoding the past is un-necessary. 1 The future is independent of the past given the present

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Markov Assumption in Speech Word Sequences Phone Sequences Part of Speech Tags Syntactic constituents Phrase sequences Discourse Acts Intonation 2

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Markov Chain The probability of a sequence can be decomposed into a probability of sequential events. 3 x1x1 x1x1 x2x2 x2x2 x3x3 x3x3

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Hidden Markov model In a Hidden Markov Model the state sequence is unobserved. Only an observation sequence is available 4 q1q1 q2q2 q3q3 x1x1 x1x1 x2x2 x2x2 x3x3 x3x3

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Hidden Markov model Observations are MFCC vectors States are phone labels Each state (phone) has an associated GMM modeling the MFCC likelihood 5 q1q1 q2q2 q3q3 x1x1 x1x1 x2x2 x2x2 x3x3 x3x3

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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 6

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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 7 q1q1 q2q2 q3q3 x1x1 x1x1 x2x2 x2x2 x3x3 x3x3

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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. 8 q1q1 q2q2 q3q3 x1x1 x1x1 x2x2 x2x2 x3x3 x3x3

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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 9

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State transitions as FSA 10 /d/ /t/ /ey/ /ax/ /ae/ /dx/

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Word FSA to phone FSA 11 /d/ /t/ /ey / /ax / /ae/ /dx/ MORE DATA /m/ /ao/ /r/

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

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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? 13

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Viterbi Decoding Dynamic Programming can make this a lot faster. Idea: Any optimal sequence between x 0 and x n must include the optimal sequence between x n and x n-1. –Based on the Markov Assumption. 14

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Viterbi Decoding 15 Probability of most likely state sequence Recovering the the optimal sequence involves storing pointers as decisions are made.

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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}, } 16 What is the most likely state sequence?

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HMM Topology for Training Rather than having one GMM per phone, it is common for acoustic models to represent each phone as 3 triphones 17 S1 S3 S2 S4 S5 /r/

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Flat Start In Flat Start training, GMM parameters are initialized to global means and variances. Viterbi is used to perform forced alignment between observations and phone sequence. –The phone sequence is derived from the lexical transcription and pronunciation model 18

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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. 19

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Flat Start In Flat Start training, GMM parameters are initialized to global means and variances. Viterbi is used to perform forced alignment between observations and phone sequence. –The phone sequence is derived from the lexical transcription and pronunciation model After alignment, retrain Acoustic Models, and repeat. 20

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What about silence? 21 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/

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Next Class Pronunciation Modeling Reading: J&M Chapter 2, Section10.5.3, 11.1, 11.2 22

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