1. Speech Processing Speech Recognition
August 19, 2005
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2. Hidden Markov Models Bonnie Dorr Christof Monz
CMSC 723: Introduction to Computational Linguistics
Lecture 5
October 6, 2004
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3. Hidden Markov Model (HMM)
HMMs allow you to estimate probabilities of unobserved events
Given plain text, which underlying parameters generated the surface
E.g., in speech recognition, the observed data is the acoustic signal and the words are the hidden parameters
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4. HMMs and their Usage
HMMs are very common in Computational Linguistics:
Speech recognition
(observed: acoustic signal, hidden: words)
Handwriting recognition
(observed: image, hidden: words)
Part-of-speech tagging
(observed: words, hidden: part-of-speech tags)
Machine translation
(observed: foreign words, hidden: words in target language)
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5. ASR Lexicon: Markov Models for pronunciation
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6. The Hidden Markov model
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7. Formal definition of HMM
States: a set of states Q = q1, q2…qN
Transition probabilities: a set of probabilities A = a01,a02,…an1,…ann.
Each aij represents P(j|i)
Observation likelihoods: a set of likelihoods B=bi(ot), probability that state i generated observation t
Special non-emitting initial and final states
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8. Pieces of the HMM
Observation likelihoods (‘b’), p(o|q), represents the acoustics of each phone, and are computed by the gaussians (“Acoustic Model”, or AM)
Transition probabilities represent the probability of different pronunciations (different sequences of phones)
States correspond to phones
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9. Pieces of the HMM
Actually, I lied when I say states correspond to phones
Actually states usually correspond to triphones
CHEESE (phones): ch iy z
CHEESE (triphones) #-ch+iy, ch-iy+z, iy-z+#
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10. Pieces of the HMM
Actually, I lied again when I said states correspond to triphones
In fact, each triphone has 3 states for beginning, middle, and end of the triphone.
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11. A real HMM
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12. Noisy Channel Model
In speech recognition you observe an acoustic signal (A=a1,…,an) and you want to determine the most likely sequence of words (W=w1,…,wn): P(W | A)
Problem: A and W are too specific for reliable counts on observed data, and are very unlikely to occur in unseen data
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13. Noisy Channel Model
Assume that the acoustic signal (A) is already segmented wrt word boundaries
P(W | A) could be computed as
Problem: Finding the most likely word corresponding to a acoustic representation depends on the context
E.g., /'pre-z&ns / could mean “presents” or “presence” depending on the context
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14. Noisy Channel Model
Given a candidate sequence W we need to compute P(W) and combine it with P(W | A)
Applying Bayes’ rule:
The denominator P(A) can be dropped, because it is constant for all W
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15. Noisy Channel in a Picture

16. Decoding The decoder combines evidence from The likelihood: P(A | W)
This can be approximated as:
The prior: P(W)
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17. Search Space
Given a word-segmented acoustic sequence list all candidates
Compute the most likely path
'bot
ik-'spen-siv
'pre-z&ns
boat
excessive
presidents
bald
expensive
presence
bold
expressive
presents
bought
inactive
press
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18. Markov Assumption
The Markov assumption states that probability of the occurrence of word wi at time t depends only on occurrence of word wi-1 at time t-1
Chain rule:
Markov assumption:
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19. The Trellis
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20. Parameters of an HMM States: A set of states S=s1,…,sn
Transition probabilities: A= a1,1,a1,2,…,an,n Each ai,j represents the probability of transitioning from state si to sj.
Emission probabilities: A set B of functions of the form bi(ot) which is the probability of observation ot being emitted by si
Initial state distribution: is the probability that si is a start state
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21. The Three Basic HMM Problems
Problem 1 (Evaluation): Given the observation sequence O=o1,…,oT and an HMM model
how do we compute the probability of O given the model?
Problem 2 (Decoding): Given the observation sequence O=o1,…,oT and an HMM model
how do we find the state sequence that best explains the observations?
Problem 3 (Learning): How do we adjust the model parameters
to maximize ?
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22. Problem 1: Probability of an Observation Sequence
What is ?
The probability of a observation sequence is the sum of the probabilities of all possible state sequences in the HMM.
Naïve computation is very expensive. Given T observations and N states, there are NT possible state sequences.
Even small HMMs, e.g. T=10 and N=10, contain 10 billion different paths
Solution to this and problem 2 is to use dynamic programming
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23. Forward Probabilities
What is the probability that, given an HMM , at time t the state is i and the partial observation o1 … ot has been generated?
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24. Forward Probabilities
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25. Forward Algorithm
Initialization:
Induction:
Termination:
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26. Forward Algorithm Complexity
In the naïve approach to solving problem 1 it takes on the order of 2T*NT computations
The forward algorithm takes on the order of N2T computations
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27. Backward Probabilities
Analogous to the forward probability, just in the other direction
What is the probability that given an HMM and given the state at time t is i, the partial observation ot+1 … oT is generated?
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28. Backward Probabilities
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29. Backward Algorithm
Initialization:
Induction:
Termination:
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30. Problem 2: Decoding
The solution to Problem 1 (Evaluation) gives us the sum of all paths through an HMM efficiently.
For Problem 2, we wan to find the path with the highest probability.
We want to find the state sequence Q=q1…qT, such that
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31. Viterbi Algorithm
Similar to computing the forward probabilities, but instead of summing over transitions from incoming states, compute the maximum
Forward:
Viterbi Recursion:
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32. Viterbi Algorithm Initialization: Induction: Termination:
Read out path:
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33. Problem 3: Learning
Up to now we’ve assumed that we know the underlying model
Often these parameters are estimated on annotated training data, which has two drawbacks:
Annotation is difficult and/or expensive
Training data is different from the current data
We want to maximize the parameters with respect to the current data, i.e., we’re looking for a model , such that
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34. Problem 3: Learning
Unfortunately, there is no known way to analytically find a global maximum, i.e., a model , such that
But it is possible to find a local maximum
Given an initial model , we can always find a model , such that
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35. Parameter Re-estimation
Use the forward-backward (or Baum-Welch) algorithm, which is a hill-climbing algorithm
Using an initial parameter instantiation, the forward-backward algorithm iteratively re-estimates the parameters and improves the probability that given observation are generated by the new parameters
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36. Parameter Re-estimation
Three parameters need to be re-estimated:
Initial state distribution:
Transition probabilities: ai,j
Emission probabilities: bi(ot)
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37. Re-estimating Transition Probabilities
What’s the probability of being in state si at time t and going to state sj, given the current model and parameters?
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38. Re-estimating Transition Probabilities
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39. Re-estimating Transition Probabilities
The intuition behind the re-estimation equation for transition probabilities is
Formally:
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40. Re-estimating Transition Probabilities
Defining
As the probability of being in state si, given the complete observation O
We can say:
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41. Review of Probabilities
Forward probability:
The probability of being in state si, given the partial observation o1,…,ot
Backward probability:
The probability of being in state si, given the partial observation ot+1,…,oT
Transition probability:
The probability of going from state si, to state sj, given the complete observation o1,…,oT
State probability:
The probability of being in state si, given the complete observation o1,…,oT
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42. Re-estimating Initial State Probabilities
Initial state distribution: is the probability that si is a start state
Re-estimation is easy:
Formally:
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43. Re-estimation of Emission Probabilities
Emission probabilities are re-estimated as
Formally:
where
Note that here is the Kronecker delta function and is not related to the in the discussion of the Viterbi algorithm!!
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44. The Updated Model Coming from we get to by the following update rules:
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45. Expectation Maximization
The forward-backward algorithm is an instance of the more general EM algorithm
The E Step: Compute the forward and backward probabilities for a give model
The M Step: Re-estimate the model parameters
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46. ASR Lexicon: Markov Models for pronunciation
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47. The Hidden Markov model
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48. Formal definition of HMM
States: a set of states Q = q1, q2…qN
Transition probabilities: a set of probabilities A = a01,a02,…an1,…ann.
Each aij represents P(j|i)
Observation likelihoods: a set of likelihoods B=bi(ot), probability that state i generated observation t
Special non-emitting initial and final states
1/18/2019

49. Pieces of the HMM
Observation likelihoods (‘b’), p(o|q), represents the acoustics of each phone, and are computed by the gaussians (“Acoustic Model”, or AM)
Transition probabilities represent the probability of different pronunciations (different sequences of phones)
States correspond to phones
1/18/2019

50. Pieces of the HMM
Actually, I lied when I say states correspond to phones
Actually states usually correspond to triphones
CHEESE (phones): ch iy z
CHEESE (triphones) #-ch+iy, ch-iy+z, iy-z+#
1/18/2019

51. Pieces of the HMM
Actually, I lied again when I said states correspond to triphones
In fact, each triphone has 3 states for beginning, middle, and end of the triphone.
1/18/2019

52. A real HMM
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53. Cross-word triphones Word-Internal Context-Dependent Models
‘OUR LIST’:
SIL AA+R AA-R L+IH L-IH+S IH-S+T S-T
Cross-Word Context-Dependent Models
SIL-AA+R AA-R+L R-L+IH L-IH+S IH-S+T S-T+SIL
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54. Summary ASR Architecture Five easy pieces of an ASR system
The Noisy Channel Model
Five easy pieces of an ASR system
Feature Extraction:
39 “MFCC” features
Acoustic Model:
Gaussians for computing p(o|q)
Lexicon/Pronunciation Model
HMM:
Next step - Decoding: how to combine these to compute words from speech!
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55. Acoustic Modeling
Given a 39d vector corresponding to the observation of one frame oi
And given a phone q we want to detect
Compute p(oi|q)
Most popular method:
GMM (Gaussian mixture models)
Other methods
MLP (multi-layer perceptron)
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56. Acoustic Modeling: MLP computes p(q|o)
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57. Gaussian Mixture Models
Also called “fully-continuous HMMs”
P(o|q) computed by a Gaussian:
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58. Gaussians for Acoustic Modeling
A Gaussian is parameterized by a mean and a variance:
Different means
P(o|q):
P(o|q) is highest here at mean
P(o|q is low here, very far from mean)
P(o|q) o
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59. Training Gaussians
A (single) Gaussian is characterized by a mean and a variance
Imagine that we had some training data in which each phone was labeled
We could just compute the mean and variance from the data:
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60. But we need 39 gaussians, not 1!
The observation o is really a vector of length 39
So need a vector of Gaussians:
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61. Actually, mixture of gaussians
Each phone is modeled by a sum of different gaussians
Hence able to model complex facts about the data
Phone A
Phone B
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62. Gaussians acoustic modeling
Summary: each phone is represented by a GMM parameterized by
M mixture weights
M mean vectors
M covariance matrices
Usually assume covariance matrix is diagonal
I.e. just keep separate variance for each cepstral feature
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63. Summary ASR Architecture Five easy pieces of an ASR system
The Noisy Channel Model
Five easy pieces of an ASR system
Feature Extraction:
39 “MFCC” features
Acoustic Model:
Gaussians for computing p(o|q)
Lexicon/Pronunciation Model
HMM:
Next time: Decoding: how to combine these to compute words from speech!
1/18/2019