1. Statistical Models for Automatic Speech Recognition
Lukáš Burget

2. Feature extraction
Preprocessing speech signal to satisfy needs of the following recognition process (dimensionality reduction, preserving only the “important” information, decorelation).
Popular features are MFCC: modification based on psycho-acoustic findings applied to short-time spectra.
For convenience, we will use one-dimensional features in most of our examples (e.g. short time energy).

3. Classifying single speech frame
unvoiced
voiced

4. Classifying single speech frame
unvoiced
voiced
Mathematically, we ask the following question: P ( v o i c e d j x )
> u n
But the value we read from probability distribution is p(x|class).
According to Bayes Rule, the above can be revritten as: p ( x j v o i c e d ) P
> u n

5. Multi-class classification
The class being correct with the highest probability is given by: a r g m x ! P ( j ) = p
silence
unvoiced
voiced
But we do not know the true distribution, …

6. Estimation of parameters
… we only see some training examples.
unvoiced
voiced
silence

7. Estimation of parameters
… we only see some training examples.
Let’s decide for some parametric model (e.g. Gaussian distribution) and estimate its parameters from the data.
unvoiced
voiced
silence

8. Maximum Likelihood Estimation
In the next part, we will use ML estimation of model parameters: ^ c l a s M L = r g m x Y 8 i 2 p ( j )
This allow as to individually estimate parameters, Θ, of each class given the data for that class. Therefore, for the convenience, we can omit the class identities in the following equations.
The models we are going to examine are:
Single Gaussian
Gaussian Mixture Model (GMM)
Hidden Markov Model
We want to solve three fundamental problems:
Evaluation of the model (computing likelihood of features given the model)
Training the model (finding ML estimates of parameters)
Finding most likely values of hidden parameters

9. Gaussian distribution (1 dimension)
Evaluation: N ( x ; 2 ) = 1 p e
ML estimates of parameters (Training): = 1 T P t x ( ) 2 = 1 T P t ( x )
No hidden variables.

10. Gaussian distribution (2 dimensions)x ; ) = 1 p 2 P j e T

11. Gaussian Mixture Model
Evaluation: p ( x j ) = P c N ;
where P c = 1

12. Gaussian Mixture Model
Evaluation: p ( x j ) = P c N ;
We can see the sum above just as a function defining the shape of the probability density function,
or we can see it as a more complicated generative probabilistic model, from which features are generated as follows:
One of Gaussian components is first randomly selected according prior probabilities Pc
Feature vector is generated form the selected Gaussian distribution
For the evaluation, however, we do not know which component generated the input vector (Identity of the component is hidden variable). Therefore, we marginalize – sum over all the components respecting their prior probabilities.
Why we want to complicate our lives with this concept:
It allows at to apply EM algorithm for GMM training
We will need this concept for HMMs

13. Training GMM –Viterbi training
Intuitive and Approximate iterative algorithm for training GMM parameters.
Using current model parameters, let Gaussians to classify data as the Gaussians were different classes (Even though the both data and all components corresponds to one class modeled by the GMM)
Re-estimate parameters of Gaussian using the data associated with to them in the previous step.
Repeat the previous two steps until the algorithm converge.

14. Training GMM – EM algorithm
Expectation Maximization is very general tool applicable in many cases were we deal with unobserved (hidden) data.
Here, we only see the result of its application to the problem of re-estimating parameters of GMM.
It guarantees to increase likelihood of training data in every iteration, however it does not guarantees to find the global optimum.
The algorithm is very similar to Viterbi training presented above. Only instead of hard decisions, it uses “soft” posterior probabilities of Gaussians (given the old model) as a weights and weight average is used to compute new mean and variance estimates. ^ ( n e w ) c = P T t 1 x ^ 2 c ( n e w ) = P T t 1 x c ( t ) = P N x ; ^ o l d 2

15. Classifying stationary sequence
unvoiced
voiced
silence
Frame independency assumption P ( X j c l a s ) = Y 8 x i 2 p

16. Modeling more general sequences: Hidden Markov Models
b1(x)
b2(x)
b3(x)
Generative model: For each frame, model moves from one state to another according to a transition probability aij and generates feature vector from probability distribution bj(.) associated with the state that was entered.
To evaluate such model, we do not se which path through the states was taken.
Let’s start with evaluating HMM for a particular state sequence.

17. a11
a22
a33
a12
a23
a34
b1(x)
b2(x)
b3(x)
P(X,S|Θ) =
b1(x1)
b1(x1)
a11
a11
b1(x2)
b1(x2)
a12
a12
b2(x3)
b2(x3)
a23
a23
b3(x4)
b3(x4)
a33
a33
b3(x5)
b3(x5)

18. Evaluating HMM for a particular state sequence
P(X,S|Θ) =
b1(x1)
b1(x1)
a11
a11
b1(x2)
b1(x2)
a12
a12
b2(x3)
b2(x3)
a23
a23
b3(x4)
b3(x4)
a33
a33
b3(x5)
b3(x5)

19. Evaluating HMM for a particular state sequence
The joint likelihood of observes sequence X and state sequence S P ( X ; S j ) = b 1 x a 2 3 4 5
can be decomposed as follows: P ( X ; S j ) =
where P ( S j ) = a 1 2 3 4
is prior probability of hidden variable – state sequence S. For GMM, the corresponding term was: Pc P ( X j S ; ) = b 1 x 2 3 4 5
is likelihood of observed sequence X, given the state sequence S. For GMM, the corresponding term was: N ( x ; c 2 )

23. .

24. .

25. .

26. Evaluating HMM (for any state sequence)
Since we do not know the underlying state sequence, we must marginalize – compute and sum likelihoods over all the possible paths P ( X j ) = S ;

32. Finding the best (Viterbi) pathsX j ) = m a x S ;

33. Training HMMs – Viterbi training
Similar to the approximate training we have already seen for GMMs
For each training utterance find Viterbi path through GMM, which associate feature frames with states.
Re-estimate state distribution using associated feature frames.
Repeat steps 1. and 2. until the algorithm converges.

34. Training HMMs using EM ° ( t ) = ^ ¹ = ^ ¾ = s ® ¯ P X j £ ® ( t ) ¯ (w ) s = P T t 1 x s ( t ) = P X j o l d ^ 2 s ( n e w ) = P T t 1 x s ( t )

35. Isolated word recognition
YES NO p ( X j Y E S ) P
> N O

36. Connected word recognition
YES
sil
sil NO

37. Phoneme based models y eh s y eh s

38. Using Language model - unigram
P(one) w ah n
one
sil
P(two) t uw
sil
two
P(three) th r iy
three

39. Using Language model - bigram
one
P(W2|W1) w ah n
sil
one
two t uw
sil
two
three th r iy
sil
three

40. Other basic ASR topics not covered by this presentation
Context dependent models
Training phoneme based models
Feature extraction
Delta parameters
De-correlation of features
Full-covariance vs. diagonal cov. modeling
Adaptation to speaker or acoustic condition
Language Modeling
LM smoothing (back-off)
Discriminative training (MMI or MPE)
and so on