Sequential Modeling with the Hidden Markov Model

Lecture 9Spoken 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.

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

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Markov Chain

• The probability of a sequence can be decomposed into a probability of sequential events.

x1 x2 x3

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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

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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

q1 q2 q3

x1 x2 x3

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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

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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

q1 q2 q3

x1 x2 x3

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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.

q1 q2 q3

x1 x2 x3

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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

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State transitions as FSA

/d/ /t//ey/ /ax/

/ae/ /dx/

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Word FSA to phone FSA

/d/ /t//ey/

/ax/

/ae/ /dx/

MORE DATA

/m/ /ao/ /r/

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Word FSA to phone FSA

/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?

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Viterbi Decoding• Dynamic Programming can make this

a lot faster.• Idea: Any optimal sequence between

x0 and xn must include the optimal sequence between xn and xn-1.– Based on the Markov Assumption.

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Viterbi Decoding• 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},}

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

S1 S3S2 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

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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.

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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.

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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/

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Next Class• Pronunciation Modeling• Reading: J&M Chapter 2,

Section10.5.3, 11.1, 11.2