Hidden Markov Models and Gaussian Mixture Models
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1 Hidden Markov Models and Gaussian Mixture Models Hiroshi Shimodaira and Steve Renals Automatic Speech Recognition ASR Lectures 4&5 23&27 January 2014 ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 1
2 Overview HMMs and GMMs Key models and algorithms for HMM acoustic models Gaussians GMMs: Gaussian mixture models HMMs: Hidden Markov models HMM algorithms Likelihood computation (forward algorithm) Most probable state sequence (Viterbi algorithm) Estimting the parameters (EM algorithm) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 2
3 Fundamental Equation of Statistical Speech Recognition If X is the sequence of acoustic feature vectors (observations) and W denotes a word sequence, the most likely word sequence W is given by W = arg max P(W X) W ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 3
4 Fundamental Equation of Statistical Speech Recognition If X is the sequence of acoustic feature vectors (observations) and W denotes a word sequence, the most likely word sequence W is given by Applying Bayes Theorem: W = arg max P(W X) W P(W X) = p(x W)P(W) p(x) p(x W)P(W) W = arg max W p(x W) }{{} Acoustic model P(W) }{{} Language model ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 3
5 Acoustic Modelling Recorded Speech Signal Analysis Decoded Text (Transcription) Hidden Markov Model Acoustic Model Training Data Lexicon Language Model Search Space ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 4
6 Hierarchical modelling of speech "No right" Utterance NO RIGHT Word n oh r ai t Subword HMM Acoustics ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 5
7 Hierarchical modelling of speech Generative Model "No right" Utterance NO RIGHT Word n oh r ai t Subword HMM Acoustics ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 5
8 Acoustic Model: Continuous Density HMM P(s 1 s 1 ) P(s 2 s 2 ) P(s 3 s 3 ) s 1 s 2 s 3 s E s I P(s 1 s I ) P(s 2 s 1 ) P(s 3 s 2 ) P(s E s 3 ) p(x s 1 ) p(x s 2 ) p(x s 3 ) x x x Probabilistic finite state automaton Paramaters λ: Transition probabilities: a kj = P(S = j S = k ) Output probability density function: b j (x) = p(x S = j ) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 6
9 Acoustic Model: Continuous Density HMM s I s 1 s 2 s 3 s E Paramaters λ: x 1 x 2 x 3 x 4 x 5 x 6 Probabilistic finite state automaton Transition probabilities: a kj = P(S = j S = k ) Output probability density function: b j (x) = p(x S = j ) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 6
10 HMM Assumptions P(s 1 s 1 ) P(s 2 s 2 ) P(s 3 s 3 ) s 1 s 2 s 3 s E s I P(s 1 s I ) P(s 2 s 1 ) P(s 3 s 2 ) P(s E s 3 ) p(x s 1 ) p(x s 2 ) p(x s 3 ) x x 1 Observation independence An acoustic observation x is conditionally independent of all other observations given the state that generated it 2 Markov process A state is conditionally independent of all other states given the previous state x ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 7
11 HMM Assumptions s(t 1) s(t) s(t+1) x(t 1) x(t) x(t + 1) 1 Observation independence An acoustic observation x is conditionally independent of all other observations given the state that generated it 2 Markov process A state is conditionally independent of all other states given the previous state ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 8
12 HMM OUTPUT DISTRIBUTION ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 9
13 Output distribution P(s 1 s 1 ) P(s 2 s 2 ) P(s 3 s 3 ) s 1 s 2 s 3 s E s I P(s 1 s I ) P(s 2 s 1 ) P(s 3 s 2 ) P(s E s 3 ) p(x s 1 ) p(x s 2 ) p(x s 3 ) x x x ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 10
14 Output distribution P(s 1 s 1 ) P(s 2 s 2 ) P(s 3 s 3 ) s 1 s 2 s 3 s E s I P(s 1 s I ) P(s 2 s 1 ) P(s 3 s 2 ) P(s E s 3 ) p(x s 1 ) p(x s 2 ) p(x s 3 ) x x x Single multivariate Gaussian with mean µ j, covariance matrix Σ j : b j (x) = p(x j ) = N (x; µ j, Σ j ) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 10
15 Output distribution P(s 1 s 1 ) P(s 2 s 2 ) P(s 3 s 3 ) s 1 s 2 s 3 s E s I P(s 1 s I ) P(s 2 s 1 ) P(s 3 s 2 ) P(s E s 3 ) p(x s 1 ) p(x s 2 ) p(x s 3 ) x x x Single multivariate Gaussian with mean µ j, covariance matrix Σ j : b j (x) = p(x j ) = N (x; µ j, Σ j ) M-component Gaussian mixture model: M b j (x) = p(x j ) = c jm N (x; µ jm, Σ jm ) m=1 ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 10
16 Output distribution P(s 1 s 1 ) P(s 2 s 2 ) P(s 3 s 3 ) s 1 s 2 s 3 s E s I P(s 1 s I ) P(s 2 s 1 ) P(s 3 s 2 ) P(s E s 3 ) p(x s 1 ) p(x s 2 ) p(x s 3 ) x x x Single multivariate Gaussian with mean µ j, covariance matrix Σ j : b j (x) = p(x j ) = N (x; µ j, Σ j ) M-component Gaussian mixture model: M b j (x) = p(x j ) = c jm N (x; µ jm, Σ jm ) m=1 ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 10
17 Background: cdf Consider a real valued random variable X Cumulative distribution function (cdf) F (x) for X : F (x) = P(X x) To obtain the probability of falling in an interval we can do the following: P(a < X b) = P(X b) P(X a) = F (b) F (a) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 11
18 Background: pdf The rate of change of the cdf gives us the probability density function (pdf), p(x): p(x) = d dx F (x) = F (x) F (x) = x p(x)dx p(x) is not the probability that X has value x. But the pdf is proportional to the probability that X lies in a small interval centred on x. Notation: p for pdf, P for probability ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 12
19 The Gaussian distribution (univariate) The Gaussian (or Normal) distribution is the most common (and easily analysed) continuous distribution It is also a reasonable model in many situations (the famous bell curve ) If a (scalar) variable has a Gaussian distribution, then it has a probability density function with this form: ( ) p(x µ, σ 2 ) = N(x; µ, σ 2 1 (x µ) 2 ) = exp 2πσ 2 2σ 2 The Gaussian is described by two parameters: the mean µ (location) the variance σ 2 (dispersion) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 13
20 Plot of Gaussian distribution Gaussians have the same shape, with the location controlled by the mean, and the spread controlled by the variance One-dimensional Gaussian with zero mean and unit variance (µ = 0, σ 2 = 1): mean=0 variance=1 pdf of Gaussian Distribution p(x m,s) x ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 14
21 Properties of the Gaussian distribution N(x; µ, σ 2 ) = ( ) 1 (x µ) 2 exp 2πσ 2 2σ pdfs of Gaussian distributions 0.35 mean=0 variance=1 0.3 mean=0 variance= p(x m,s) mean=0 variance= x ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 15
22 Parameter estimation Estimate mean and variance parameters of a Gaussian from data x (1), x (2),..., x (N) Use sample mean and sample variance estimates: µ = 1 N σ 2 = 1 N N n=1 x (n) (sample mean) N (x (n) µ) 2 (sample variance) n=1 ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 16
23 Exercise maximum likelihood estimation (MLE) Consider the log likelihood of a set of N training data points {x (1),..., x (N) } being generated by a Gaussian with mean µ and variance σ 2 : L = ln p({x (1),..., x (n) } µ, σ 2 ) = 1 2 = 1 2σ 2 N n=1 ( (x (n) µ) 2 σ 2 ln σ 2 ln(2π) N (x (n) µ) 2 N 2 ln σ2 N 2 ln(2π) n=1 By maximising the the log likelihood function with respect to µ show that the maximum likelihood estimate for the mean is indeed the sample mean: µ ML = 1 N x (n). N n=1 ) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 17
24 The multivariate Gaussian distribution The d-dimensional vector x = (x 1,..., x d ) T follows a multivariate Gaussian (or normal) distribution if it has a probability density function of the following form: ( 1 p(x µ, Σ) = (2π) d/2 exp 1 ) Σ 1/2 2 (x µ)t Σ 1 (x µ) The pdf is parameterized by the mean vector µ = (µ 1,..., µ d ) T σ σ 1d and the covariance matrix Σ = σ d1... σ dd ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 18
25 The multivariate Gaussian distribution The d-dimensional vector x = (x 1,..., x d ) T follows a multivariate Gaussian (or normal) distribution if it has a probability density function of the following form: ( 1 p(x µ, Σ) = (2π) d/2 exp 1 ) Σ 1/2 2 (x µ)t Σ 1 (x µ) The pdf is parameterized by the mean vector µ = (µ 1,..., µ d ) T σ σ 1d and the covariance matrix Σ = σ d1... σ dd The 1-dimensional Gaussian is a special case of this pdf ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 18
26 The multivariate Gaussian distribution The d-dimensional vector x = (x 1,..., x d ) T follows a multivariate Gaussian (or normal) distribution if it has a probability density function of the following form: ( 1 p(x µ, Σ) = (2π) d/2 exp 1 ) Σ 1/2 2 (x µ)t Σ 1 (x µ) The pdf is parameterized by the mean vector µ = (µ 1,..., µ d ) T σ σ 1d and the covariance matrix Σ = σ d1... σ dd The 1-dimensional Gaussian is a special case of this pdf The argument to the exponential 0.5(x µ) T Σ 1 (x µ) is referred to as a quadratic form. ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 18
27 Covariance matrix The mean vector µ is the expectation of x: µ = E[x] ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 19
28 Covariance matrix The mean vector µ is the expectation of x: µ = E[x] The covariance matrix Σ is the expectation of the deviation of x from the mean: Σ = E[(x µ)(x µ) T ] ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 19
29 Covariance matrix The mean vector µ is the expectation of x: µ = E[x] The covariance matrix Σ is the expectation of the deviation of x from the mean: Σ = E[(x µ)(x µ) T ] Σ is a d d symmetric matrix: σ ij = E[(x i µ i )(x j µ j )] = E[(x j µ j )(x i µ i )] = σ ji ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 19
30 Covariance matrix The mean vector µ is the expectation of x: µ = E[x] The covariance matrix Σ is the expectation of the deviation of x from the mean: Σ = E[(x µ)(x µ) T ] Σ is a d d symmetric matrix: σ ij = E[(x i µ i )(x j µ j )] = E[(x j µ j )(x i µ i )] = σ ji The sign of the covariance helps to determine the relationship between two components: ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 19
31 Covariance matrix The mean vector µ is the expectation of x: µ = E[x] The covariance matrix Σ is the expectation of the deviation of x from the mean: Σ = E[(x µ)(x µ) T ] Σ is a d d symmetric matrix: σ ij = E[(x i µ i )(x j µ j )] = E[(x j µ j )(x i µ i )] = σ ji The sign of the covariance helps to determine the relationship between two components: If x j is large when x i is large, then (x i µ i )(x j µ j ) will tend to be positive; ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 19
32 Covariance matrix The mean vector µ is the expectation of x: µ = E[x] The covariance matrix Σ is the expectation of the deviation of x from the mean: Σ = E[(x µ)(x µ) T ] Σ is a d d symmetric matrix: σ ij = E[(x i µ i )(x j µ j )] = E[(x j µ j )(x i µ i )] = σ ji The sign of the covariance helps to determine the relationship between two components: If x j is large when x i is large, then (x i µ i )(x j µ j ) will tend to be positive; If x j is small when x i is large, then (x i µ i )(x j µ j ) will tend to be negative. ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 19
33 Spherical Gaussian Surface plot of p(x 1, x 2 ) 2 Contour plot of p(x 1, x 2 ) p(x 1, x 2 ) x x µ = ( 0 0 x 1 ) Σ = x 1 ( ) ρ 12 = 0 ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 20
34 Diagonal Covariance Gaussian Surface plot of p(x 1, x 2 ) 4 Contour plot of p(x 1, x 2 ) p(x 1, x 2 ) x x 2 2 µ = 4 4 ( ) x 1 Σ = x 1 ( ) ρ 12 = 0 ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 21
35 Full covariance Gaussian Surface plot of p(x 1, x 2 ) 4 Contour plot of p(x 1, x 2 ) p(x 1, x 2 ) x x 2 µ = 2 4 ( ) 2 1 x Σ = ( x 1 ) ρ 12 = 0.5 ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 22
36 Parameter estimation of a multivariate Gaussian distribution It is possible to show that the mean vector ˆµ and covariance matrix ˆΣ that maximize the likelihood of the training data are given by: ˆµ = 1 N ˆΣ = 1 N N n=1 x (n) N (x (n) ˆµ)(x (n) ˆµ) T n=1 The mean of the distribution is estimated by the sample mean and the covariance by the sample covariance ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 23
37 Example data 10 5 X X1 ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 24
38 Maximum likelihood fit to a Gaussian 10 5 X X1 ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 25
39 Data in clusters (example 1) µ 1 = [0 0] T µ 2 = [1 1] T Σ 1 = Σ 2 = 0.2 I ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 26
40 Example 1 fit by a Gaussian µ 1 = [0 0] T µ 2 = [1 1] T Σ 1 = Σ 2 = 0.2 I ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 27
41 k-means clustering k-means is an automatic procedure for clustering unlabelled data Requires a prespecified number of clusters Clustering algorithm chooses a set of clusters with the minimum within-cluster variance Guaranteed to converge (eventually) Clustering solution is dependent on the initialisation ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 28
42 k-means example: data set (4,13) 10 (2,9) (7,8) 5 (6,6) (7,6) (4,5) (5,4) (8,4) (10,5) 0 (1,2) (5,2) (1,1) (3,1) (10,0) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 29
43 k-means example: initialization (4,13) 10 (2,9) (7,8) 5 (6,6) (7,6) (4,5) (5,4) (8,4) (10,5) 0 (1,2) (5,2) (1,1) (3,1) (10,0) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 30
44 k-means example: iteration 1 (assign points to clusters) (4,13) 10 (2,9) (7,8) 5 (6,6) (7,6) (4,5) (5,4) (8,4) (10,5) 0 (1,2) (5,2) (1,1) (3,1) (10,0) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 31
45 k-means example: iteration 1 (recompute centres) (4,13) 10 (2,9) (4.33, 10) (7,8) 5 0 (6,6) (7,6) (4,5) (10,5) (5,4) (8,4) (8.75,3.75) (3.57, 3) (1,2) (5,2) (1,1) (3,1) (10,0) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 32
46 k-means example: iteration 2 (assign points to clusters) (4,13) 10 (2,9) (4.33, 10) (7,8) 5 0 (7,6) (6,6) (4,5) (10,5) (5,4) (8,4) (8.75,3.75) (3.57, 3) (1,2) (5,2) (1,1) (3,1) (10,0) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 33
47 k-means example: iteration 2 (recompute centres) (4,13) 10 (2,9) (4.33, 10) (7,8) 5 (6,6) (7,6) (4,5) (5,4) (8,4) (10,5) (8.2,4.2) 0 (1,2) (1,1) (3.17, 2.5) (5,2) (3,1) (10,0) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 34
48 k-means example: iteration 3 (assign points to clusters) (4,13) 10 (2,9) (4.33, 10) (7,8) 5 (4,5) (6,6) (5,4) (7,6) (8,4) (10,5) (8.2,4.2) 0 (1,2) (1,1) No changes, so converged (3.17, 2.5) (5,2) (3,1) (10,0) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 35
49 Mixture model A more flexible form of density estimation is made up of a linear combination of component densities: p(x) = M p(x j )P( j ) j=1 This is called a mixture model or a mixture density p(x j ): component densities P( j ): mixing parameters Generative model: 1 Choose a mixture component based on P( j ) 2 Generate a data point x from the chosen component using p(x j ) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 36
50 Component occupation probability We can apply Bayes theorem: P( j x) = p(x j )P( j ) p(x) = p(x j )P( j ) M j=1 p(x j )P( j ) The posterior probabilities P( j x) give the probability that component j was responsible for generating data point x The P( j x)s are called the component occupation probabilities (or sometimes called the responsibilities) Since they are posterior probabilities: M P( j x) = 1 j=1 ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 37
51 Parameter estimation If we knew which mixture component was responsible for a data point: we would be able to assign each point unambiguously to a mixture component and we could estimate the mean for each component Gaussian as the sample mean (just like k-means clustering) and we could estimate the covariance as the sample covariance But we don t know which mixture component a data point comes from... Maybe we could use the component occupation probabilities P( j x)? ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 38
52 Gaussian mixture model The most important mixture model is the Gaussian Mixture Model (GMM), where the component densities are Gaussians Consider a GMM, where each component Gaussian N j (x; µ j, Σ j ) has mean µ j and a spherical covariance Σ j = σ 2 j I p(x) = M P( j )p(x j ) = j=1 p(x) M P( j )N j (x; µ j, σj 2 I) j=1 P(1) P(2) P(M) p(x 1) p(x 2) p(x M) x 1 x 2 x d ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 39
53 GMM Parameter estimation when we know which component generated the data Define the indicator variable z jn = 1 if component j generated component x (n) (and 0 otherwise) If z jn wasn t hidden then we could count the number of observed data points generated by j: N N j = n=1 And estimate the mean, variance and mixing parameters as: ˆµ j = ˆσ 2 j = ˆP( j ) = 1 N z jn n z jnx (n) N j n z jn x (n) ˆµ j 2 N j z jn = N j N n ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 40
54 Soft assignment Estimate soft counts based on the component occupation probabilities P( j x (n) ): N Nj = P( j x (n) ) n=1 We can imagine assigning data points to component j weighted by the component occupation probability P( j x (n) ) So we could imagine estimating the mean, variance and prior probabilities as: n ˆµ j = P( j x(n) )x (n) n n P( j = P( j x(n) )x (n) x(n) ) Nj n ˆσ j 2 = P( j x(n) ) x (n) µ j 2 n n P( j = P( j x(n) ) x (n) µ j 2 x(n) ) Nj ˆP( j ) = 1 N n P( j x (n) ) = N j N ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 41
55 EM algorithm Problem! Recall that: P( j x) = p(x j )P( j ) p(x) p(x j )P( j ) = M j=1 p(x j )P( j ) We need to know p(x j ) and P( j ) to estimate the parameters of p( j x), and to estimate P( j )... Solution: an iterative algorithm where each iteration has two parts: Compute the component occupation probabilities P( j x) using the current estimates of the GMM parameters (means, variances, mixing parameters) (E-step) Computer the GMM parameters using the current estimates of the component occupation probabilities (M-step) Starting from some initialization (e.g. using k-means for the means) these steps are alternated until convergence This is called the EM Algorithm and can be shown to maximize the likelihood ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 42
56 Maximum likelihood parameter estimation The likelihood of a data set X = {x (1), x (2),..., x (N) } is given by: N N M L = p(x (n) ) = p(x (n) j )P( j ) n=1 n=1 j=1 We can regard the negative log likelihood as an error function: N E = ln L = ln p(x (n) ) n=1 N M = ln p(x (n) j )P( j ) n=1 j=1 Considering the derivatives of E with respect to the parameters, gives expressions like the previous slide ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 43
57 Example 1 fit using a GMM ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 44
58 Example 1 fit using a GMM Fitted with a two component GMM using EM ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 44
59 Peakily distributed data (Example 2) µ 1 = µ 2 = [0 0] T Σ 1 = 0.1I Σ 2 = 2I ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 45
60 Example 2 fit by a Gaussian µ 1 = µ 2 = [0 0] T Σ 1 = 0.1I Σ 2 = 2I ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 46
61 Example 2 fit by a GMM ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 47
62 Example 2 fit by a GMM Fitted with a two component GMM using EM ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 47
63 Example 2: component Gaussians ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 48
64 Comments on GMMs GMMs trained using the EM algorithm are able to self organize to fit a data set Individual components take responsibility for parts of the data set (probabilistically) Soft assignment to components not hard assignment soft clustering GMMs scale very well, e.g.: large speech recognition systems can have 30,000 GMMs, each with 32 components: sometimes 1 million Gaussian components!! And the parameters all estimated from (a lot of) data by EM ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 49
65 Back to HMMs... P(s 1 s 1 ) P(s 2 s 2 ) P(s 3 s 3 ) s 1 s 2 s 3 s E s I P(s 1 s I ) P(s 2 s 1 ) P(s 3 s 2 ) P(s E s 3 ) p(x s 1 ) p(x s 2 ) p(x s 3 ) x Output distribution: Single multivariate Gaussian with mean µ j, covariance matrix Σ j : b j (x) = p(x S = j ) = N (x; µ j, Σ j ) M-component Gaussian mixture model: M b j (x) = p(x S = j ) = c jm N (x; µ jm, Σ jm ) x x m=1 ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 50
66 The three problems of HMMs Working with HMMs requires the solution of three problems: 1 Likelihood Determine the overall likelihood of an observation sequence X = (x 1,..., x t,..., x T ) being generated by an HMM. (NB: x t is used to denote x (t) hereafter) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 51
67 The three problems of HMMs Working with HMMs requires the solution of three problems: 1 Likelihood Determine the overall likelihood of an observation sequence X = (x 1,..., x t,..., x T ) being generated by an HMM. (NB: x t is used to denote x (t) hereafter) 2 Decoding Given an observation sequence and an HMM, determine the most probable hidden state sequence ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 51
68 The three problems of HMMs Working with HMMs requires the solution of three problems: 1 Likelihood Determine the overall likelihood of an observation sequence X = (x 1,..., x t,..., x T ) being generated by an HMM. (NB: x t is used to denote x (t) hereafter) 2 Decoding Given an observation sequence and an HMM, determine the most probable hidden state sequence 3 Training Given an observation sequence and an HMM, learn the best HMM parameters λ = {{a jk }, {b j ()}} ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 51
69 1. Likelihood: how to calculate? S 4 33 a34 a a 22 a 11 states a a S 23 S S a 1 01 trellis S time x 2 x 3 x 4 x 5 x 6 x 7 observations x 1 P(X, path j Λ) = P(X path j, Λ)P(path j Λ) = P(X s 0 s 1 s 1 s 1 s 2 s 2 s 3 s 3 s 4, Λ)P(s 0 s 1 s 1 s 1 s 2 s 2 s 3 s 3 s 4 Λ) = b 1 (x 1 )b 1 (x 2 )b 1 (x 3 )b 2 (x 4 )b 2 (x 5 )b 3 (x 6 )b 3 (x 7 )a 01 a 11 a 11 a 12 a 22 a 23 a 33 a 34 P(X Λ) = {path j } P(X, path j Λ) max path j P(X, path j Λ) forward(backward) algorithm Viterbi algorithm ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 52
70 1. Likelihood: The Forward algorithm Goal: determine p(x λ) Sum over all possible state sequences s 1 s 2... s T that could result in the observation sequence X ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 53
71 1. Likelihood: The Forward algorithm Goal: determine p(x λ) Sum over all possible state sequences s 1 s 2... s T that could result in the observation sequence X Rather than enumerating each sequence, compute the probabilities recursively (exploiting the Markov assumption) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 53
72 1. Likelihood: The Forward algorithm Goal: determine p(x λ) Sum over all possible state sequences s 1 s 2... s T that could result in the observation sequence X Rather than enumerating each sequence, compute the probabilities recursively (exploiting the Markov assumption) Hown many paths calculations in p(x λ)? N N N }{{} T times = N T N : number of HMM states T : length of observation e.g. N = 3, T = ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 53
73 1. Likelihood: The Forward algorithm Goal: determine p(x λ) Sum over all possible state sequences s 1 s 2... s T that could result in the observation sequence X Rather than enumerating each sequence, compute the probabilities recursively (exploiting the Markov assumption) Hown many paths calculations in p(x λ)? N N N }{{} T times = N T N : number of HMM states T : length of observation e.g. N = 3, T = Computation complexity for multiplication: O(2T N T ) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 53
74 1. Likelihood: The Forward algorithm Goal: determine p(x λ) Sum over all possible state sequences s 1 s 2... s T that could result in the observation sequence X Rather than enumerating each sequence, compute the probabilities recursively (exploiting the Markov assumption) Hown many paths calculations in p(x λ)? N N N }{{} T times = N T N : number of HMM states T : length of observation e.g. N = 3, T = Computation complexity for multiplication: O(2T N T ) The Forward algorithm reduces this to O(TN 2 ) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 53
75 Recursive algorithms on HMMs Visualize the problem as a state-time trellis t-1 t t+1 i i i j j j k k k ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 54
76 1. Likelihood: The Forward algorithm Goal: determine p(x λ) Sum over all possible state sequences s 1 s 2... s T that could result in the observation sequence X Rather than enumerating each sequence, compute the probabilities recursively (exploiting the Markov assumption) Forward probability, α t ( j ): the probability of observing the observation sequence x 1... x t and being in state j at time t: α t ( j ) = p(x 1,..., x t, S(t) = j λ) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 55
77 1. Likelihood: The Forward recursion Initialization α 0 (s I ) = 1 α 0 ( j ) = 0 if j s I ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 56
78 1. Likelihood: The Forward recursion Initialization α 0 (s I ) = 1 α 0 ( j ) = 0 if j s I Recursion α t ( j ) = N α t 1 ( i )a ij b j (x t ) i=1 1 j N, 1 t T ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 56
79 1. Likelihood: The Forward recursion Initialization α 0 (s I ) = 1 α 0 ( j ) = 0 if j s I Recursion α t ( j ) = Termination N α t 1 ( i )a ij b j (x t ) i=1 p(x λ) = α T (s E ) = 1 j N, 1 t T N α T ( i )a ie i=1 s I : initial state, s E : final state ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 56
80 1. Likelihood: Forward Recursion α t ( j ) = p(x 1,..., x t, S(t) = j λ) = i α ( i ) t 1 j α ( j ) t 1 t 1 a ij a a jj kj Σ j α ( j ) t t i N α t 1 ( i )a ij b j (x t ) i=1 t+1 i j k α ( k ) t 1 k k ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 57
81 Viterbi approximation Instead of summing over all possible state sequences, just consider the most likely ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 58
82 Viterbi approximation Instead of summing over all possible state sequences, just consider the most likely Achieve this by changing the summation to a maximisation in the recursion: V t ( j ) = max V t 1 ( i )a ij b j (x t ) i ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 58
83 Viterbi approximation Instead of summing over all possible state sequences, just consider the most likely Achieve this by changing the summation to a maximisation in the recursion: V t ( j ) = max V t 1 ( i )a ij b j (x t ) i Changing the recursion in this way gives the likelihood of the most probable path ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 58
84 Viterbi approximation Instead of summing over all possible state sequences, just consider the most likely Achieve this by changing the summation to a maximisation in the recursion: V t ( j ) = max V t 1 ( i )a ij b j (x t ) i Changing the recursion in this way gives the likelihood of the most probable path We need to keep track of the states that make up this path by keeping a sequence of backpointers to enable a Viterbi backtrace: the backpointer for each state at each time indicates the previous state on the most probable path ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 58
85 Viterbi Recursion V t ( j ) = max V t 1 ( i )a ij b j (x t ) i Likelihood of the most probable path i V ( i ) t 1 j V ( j ) t 1 t 1 a ij ajj max j akj V ( j ) t t t+1 i i j k V ( k ) t 1 k k ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 59
86 Viterbi Recursion Backpointers to the previous state on the most probable path i i a V ij t 1( i ) bt t ( j )= i ajj j j V ( j ) akj V ( j ) t 1 t 1 t t+1 t i j k V ( k ) t 1 k k ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 60
87 2. Decoding: The Viterbi algorithm Initialization V 0 ( i ) = 1 V 0 ( j ) = 0 bt 0 ( j ) = 0 if j i ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 61
88 2. Decoding: The Viterbi algorithm Initialization V 0 ( i ) = 1 V 0 ( j ) = 0 bt 0 ( j ) = 0 if j i Recursion V t ( j ) = N max i=1 V t 1( i )a ij b j (x t ) bt t ( j ) = arg N max i=1 V t 1( i )a ij b j (x t ) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 61
89 2. Decoding: The Viterbi algorithm Initialization V 0 ( i ) = 1 V 0 ( j ) = 0 bt 0 ( j ) = 0 if j i Recursion Termination V t ( j ) = N max i=1 V t 1( i )a ij b j (x t ) bt t ( j ) = arg N max i=1 V t 1( i )a ij b j (x t ) P = V T (s E ) = N max i=1 V T ( i )a ie s T = bt T (q E ) = arg N max i=1 V T ( i )a ie ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 61
90 Viterbi Backtrace Backtrace to find the state sequence of the most probable path t 1 t bt t ( i )= j t+1 i i i V ( i ) t 1 j V ( j ) t 1 j j k V ( k ) t 1 k k bt ( k )= i t+1 ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 62
91 3. Training: Forward-Backward algorithm Goal: Efficiently estimate the parameters of an HMM λ from an observation sequence ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 63
92 3. Training: Forward-Backward algorithm Goal: Efficiently estimate the parameters of an HMM λ from an observation sequence Assume single Gaussian output probability distribution b j (x) = p(x j ) = N (x; µ j, Σ j ) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 63
93 3. Training: Forward-Backward algorithm Goal: Efficiently estimate the parameters of an HMM λ from an observation sequence Assume single Gaussian output probability distribution Parameters λ: Transition probabilities a ij : b j (x) = p(x j ) = N (x; µ j, Σ j ) a ij = 1 Gaussian parameters for state j : mean vector µ j ; covariance matrix Σ j j ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 63
94 Viterbi Training If we knew the state-time alignment, then each observation feature vector could be assigned to a specific state ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 64
95 Viterbi Training If we knew the state-time alignment, then each observation feature vector could be assigned to a specific state A state-time alignment can be obtained using the most probable path obtained by Viterbi decoding ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 64
96 Viterbi Training If we knew the state-time alignment, then each observation feature vector could be assigned to a specific state A state-time alignment can be obtained using the most probable path obtained by Viterbi decoding Maximum likelihood estimate of a ij, if C( i j ) is the count of transitions from i to j â ij = C( i j ) k C( i k ) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 64
97 Viterbi Training If we knew the state-time alignment, then each observation feature vector could be assigned to a specific state A state-time alignment can be obtained using the most probable path obtained by Viterbi decoding Maximum likelihood estimate of a ij, if C( i j ) is the count of transitions from i to j â ij = C( i j ) k C( i k ) Likewise if Z j is the set of observed acoustic feature vectors assigned to state j, we can use the standard maximum likelihood estimates for the mean and the covariance: x Z ˆµ j = j x Z j x Z ˆΣ j = j (x ˆµ j )(x ˆµ j ) T Z j ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 64
98 EM Algorithm Viterbi training is an approximation we would like to consider all possible paths ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 65
99 EM Algorithm Viterbi training is an approximation we would like to consider all possible paths In this case rather than having a hard state-time alignment we estimate a probability ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 65
100 EM Algorithm Viterbi training is an approximation we would like to consider all possible paths In this case rather than having a hard state-time alignment we estimate a probability State occupation probability: The probability γ t ( j ) of occupying state j at time t given the sequence of observations. Compare with component occupation probability in a GMM ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 65
101 EM Algorithm Viterbi training is an approximation we would like to consider all possible paths In this case rather than having a hard state-time alignment we estimate a probability State occupation probability: The probability γ t ( j ) of occupying state j at time t given the sequence of observations. Compare with component occupation probability in a GMM We can use this for an iterative algorithm for HMM training: the EM algorithm (whose adaption to HMM is called Baum-Welch algorithm ) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 65
102 EM Algorithm Viterbi training is an approximation we would like to consider all possible paths In this case rather than having a hard state-time alignment we estimate a probability State occupation probability: The probability γ t ( j ) of occupying state j at time t given the sequence of observations. Compare with component occupation probability in a GMM We can use this for an iterative algorithm for HMM training: the EM algorithm (whose adaption to HMM is called Baum-Welch algorithm ) Each iteration has two steps: ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 65
103 EM Algorithm Viterbi training is an approximation we would like to consider all possible paths In this case rather than having a hard state-time alignment we estimate a probability State occupation probability: The probability γ t ( j ) of occupying state j at time t given the sequence of observations. Compare with component occupation probability in a GMM We can use this for an iterative algorithm for HMM training: the EM algorithm (whose adaption to HMM is called Baum-Welch algorithm ) Each iteration has two steps: E-step estimate the state occupation probabilities (Expectation) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 65
104 EM Algorithm Viterbi training is an approximation we would like to consider all possible paths In this case rather than having a hard state-time alignment we estimate a probability State occupation probability: The probability γ t ( j ) of occupying state j at time t given the sequence of observations. Compare with component occupation probability in a GMM We can use this for an iterative algorithm for HMM training: the EM algorithm (whose adaption to HMM is called Baum-Welch algorithm ) Each iteration has two steps: E-step estimate the state occupation probabilities (Expectation) M-step re-estimate the HMM parameters based on the estimated state occupation probabilities (Maximisation) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 65
105 Backward probabilities To estimate the state occupation probabilities it is useful to define (recursively) another set of probabilities the Backward probabilities β t ( j ) = p(x t+1,..., x T S(t) = j, λ) The probability of future observations given a the HMM is in state j at time t ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 66
106 Backward probabilities To estimate the state occupation probabilities it is useful to define (recursively) another set of probabilities the Backward probabilities β t ( j ) = p(x t+1,..., x T S(t) = j, λ) The probability of future observations given a the HMM is in state j at time t These can be recursively computed (going backwards in time) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 66
107 Backward probabilities To estimate the state occupation probabilities it is useful to define (recursively) another set of probabilities the Backward probabilities β t ( j ) = p(x t+1,..., x T S(t) = j, λ) The probability of future observations given a the HMM is in state j at time t These can be recursively computed (going backwards in time) Initialisation β T ( i ) = a ie ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 66
108 Backward probabilities To estimate the state occupation probabilities it is useful to define (recursively) another set of probabilities the Backward probabilities β t ( j ) = p(x t+1,..., x T S(t) = j, λ) The probability of future observations given a the HMM is in state j at time t These can be recursively computed (going backwards in time) Initialisation β T ( i ) = a ie Recursion β t ( i ) = N a ij b j (x t+1 )β t+1 ( j ) for t = T 1,..., 1 j=1 ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 66
109 Backward probabilities To estimate the state occupation probabilities it is useful to define (recursively) another set of probabilities the Backward probabilities β t ( j ) = p(x t+1,..., x T S(t) = j, λ) The probability of future observations given a the HMM is in state j at time t These can be recursively computed (going backwards in time) Initialisation β T ( i ) = a ie Recursion β t ( i ) = Termination N a ij b j (x t+1 )β t+1 ( j ) for t = T 1,..., 1 j=1 p(x λ) = β 0 ( I ) = N a Ij b j (x 1 )β 1 ( j ) = α T (s E ) j=1 ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 66
110 Backward Recursion N β t ( j ) = p(x t+1,..., x T S(t) = j, λ) = a ij b j (x t+1 )β t+1 ( j ) t 1 i j j β ( j ) t t i Σ a a a ji jk jj j=1 t+1 i β ( i ) t+1 j β ( j ) t+1 k k k β ( k) t+1 ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 67
111 State Occupation Probability The state occupation probability γ t ( j ) is the probability of occupying state j at time t given the sequence of observations ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 68
112 State Occupation Probability The state occupation probability γ t ( j ) is the probability of occupying state j at time t given the sequence of observations Express in terms of the forward and backward probabilities: 1 γ t ( j ) = P(S(t) = j X, λ) = α T (s E ) α t( j )β t ( j ) recalling that p(x λ) = α T (s E ) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 68
113 State Occupation Probability The state occupation probability γ t ( j ) is the probability of occupying state j at time t given the sequence of observations Express in terms of the forward and backward probabilities: 1 γ t ( j ) = P(S(t) = j X, λ) = α T (s E ) α t( j )β t ( j ) recalling that p(x λ) = α T (s E ) Since α t ( j )β t ( j ) = p(x 1,..., x t, S(t) = j λ) p(x t+1,..., x T S(t) = j, λ) = p(x 1,..., x t, x t+1,..., x T, S(t) = j λ) = p(x, S(t) = j λ) P(S(t) = j X, λ) = p(x, S(t) = j λ) p(x λ) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 68
114 Re-estimation of Gaussian parameters The sum of state occupation probabilities through time for a state, may be regarded as a soft count ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 69
115 Re-estimation of Gaussian parameters The sum of state occupation probabilities through time for a state, may be regarded as a soft count We can use this soft alignment to re-estimate the HMM parameters: ˆµ j = ˆΣ j = T t=1 γ t( j )x t T t=1 γ t( j ) T t=1 γ t( j )(x t ˆµ j )(x ˆµ j ) T T t=1 γ t( j ) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 69
116 Re-estimation of transition probabilities Similarly to the state occupation probability, we can estimate ξ t ( i, j ), the probability of being in i at time t and j at t + 1, given the observations: ξ t ( i, j ) = P(S(t) = i, S(t + 1) = j X, λ) P(S(t) = i, S(t + 1) = j, X λ) = p(x Λ) = α t( i )a ij b j (x t+1 )β t+1 ( j ) α T (s E ) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 70
117 Re-estimation of transition probabilities Similarly to the state occupation probability, we can estimate ξ t ( i, j ), the probability of being in i at time t and j at t + 1, given the observations: ξ t ( i, j ) = P(S(t) = i, S(t + 1) = j X, λ) P(S(t) = i, S(t + 1) = j, X λ) = p(x Λ) = α t( i )a ij b j (x t+1 )β t+1 ( j ) α T (s E ) We can use this to re-estimate the transition probabilities â ij = T t=1 ξ t( i, j ) N T k=1 t=1 ξ t( i, k ) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 70
118 Pulling it all together Iterative estimation of HMM parameters using the EM algorithm. At each iteration E step For all time-state pairs 1 Recursively compute the forward probabilities α t ( j ) and backward probabilities β t ( j ) 2 Compute the state occupation probabilities γ t ( j ) and ξ t ( i, j ) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 71
119 Pulling it all together Iterative estimation of HMM parameters using the EM algorithm. At each iteration E step For all time-state pairs 1 Recursively compute the forward probabilities α t ( j ) and backward probabilities β t ( j ) 2 Compute the state occupation probabilities γ t ( j ) and ξ t ( i, j ) M step Based on the estimated state occupation probabilities re-estimate the HMM parameters: mean vectors µ j, covariance matrices Σ j and transition probabilities a ij ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 71
120 Pulling it all together Iterative estimation of HMM parameters using the EM algorithm. At each iteration E step For all time-state pairs 1 Recursively compute the forward probabilities α t ( j ) and backward probabilities β t ( j ) 2 Compute the state occupation probabilities γ t ( j ) and ξ t ( i, j ) M step Based on the estimated state occupation probabilities re-estimate the HMM parameters: mean vectors µ j, covariance matrices Σ j and transition probabilities a ij The application of the EM algorithm to HMM training is sometimes called the Forward-Backward algorithm ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 71
121 Extension to a corpus of utterances We usually train from a large corpus of R utterances If x r t is the t th frame of the r th utterance X r then we can compute the probabilities α r t( j ), β r t ( j ), γ r t ( j ) and ξ r t ( i, j ) as before The re-estimates are as before, except we must sum over the R utterances, eg: R T r=1 t=1 ˆµ j = γr t ( j )x r t R T r=1 t=1 γr t ( j ) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 72
122 Extension to Gaussian mixture model (GMM) The assumption of a Gaussian distribution at each state is very strong; in practice the acoustic feature vectors associated with a state may be strongly non-gaussian ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 73
123 Extension to Gaussian mixture model (GMM) The assumption of a Gaussian distribution at each state is very strong; in practice the acoustic feature vectors associated with a state may be strongly non-gaussian In this case an M-component Gaussian mixture model is an appropriate density function: b j (x) = p(x j ) = M c jm N (x; µ jm, Σ jm ) m=1 Given enough components, this family of functions can model any distribution. ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 73
124 Extension to Gaussian mixture model (GMM) The assumption of a Gaussian distribution at each state is very strong; in practice the acoustic feature vectors associated with a state may be strongly non-gaussian In this case an M-component Gaussian mixture model is an appropriate density function: b j (x) = p(x j ) = M c jm N (x; µ jm, Σ jm ) m=1 Given enough components, this family of functions can model any distribution. Train using the EM algorithm, in which the component estimation probabilities are estimated in the E-step ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 73
125 EM training of HMM/GMM Rather than estimating the state-time alignment, we estimate the component/state-time alignment, and component-state occupation probabilities γ t ( j, m): the probability of occupying mixture component m of state j at time t ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 74
126 EM training of HMM/GMM Rather than estimating the state-time alignment, we estimate the component/state-time alignment, and component-state occupation probabilities γ t ( j, m): the probability of occupying mixture component m of state j at time t We can thus re-estimate the mean of mixture component m of state j as follows ˆµ jm = T t=1 γ t( j, m)x t T t=1 γ t( j, m) And likewise for the covariance matrices (mixture models often use diagonal covariance matrices) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 74
127 EM training of HMM/GMM Rather than estimating the state-time alignment, we estimate the component/state-time alignment, and component-state occupation probabilities γ t ( j, m): the probability of occupying mixture component m of state j at time t We can thus re-estimate the mean of mixture component m of state j as follows ˆµ jm = T t=1 γ t( j, m)x t T t=1 γ t( j, m) And likewise for the covariance matrices (mixture models often use diagonal covariance matrices) The mixture coefficients are re-estimated in a similar way to transition probabilities: T t=1 ĉ jm = γ t( j, m) M T l=1 t=1 γ t( j, l) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 74
128 Doing the computation The forward, backward and Viterbi recursions result in a long sequence of probabilities being multiplied This can cause floating point underflow problems In practice computations are performed in the log domain (in which multiplies become adds) Working in the log domain also avoids needing to perform the exponentiation when computing Gaussians ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 75
129 A note on HMM topology left to right model a 11 a a 22 a a 33 parallel path left to right model a 11 a 12 a a 22 a 23 a a 33 a 34 a a 44 a a 55 ergodic model a 11 a 12 a 13 a 14 a 15 a 21 a 22 a 23 a 24 a 25 a 31 a 32 a 33 a 34 a 35 a 41 a 42 a 43 a 44 a 45 a 51 a 52 a 53 a 54 a 55 Speech recognition: Speaker recognition: left-to-right HMM with 3 5 states ergodic HMM ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 76
130 A note on HMM emission probabilities a 11 a 22 a 33 1 a 01 b (x) a 12 b (x) 2 a b (x) 3 a emission pdfs Emission prob. Continuous (density) HMM continuous density GMM, NN/DNN Discrete (probability) HMM discrete probability VQ Semi-continuous HMM continuous density tied mixture (tied-mixture HMM) ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 77
131 Summary: HMMs HMMs provide a generative model for statistical speech recognition Three key problems 1 Computing the overall likelihood: the Forward algorithm 2 Decoding the most likely state sequence: the Viterbi algorithm 3 Estimating the most likely parameters: the EM (Forward-Backward) algorithm Solutions to these problems are tractable due to the two key HMM assumptions 1 Conditional independence of observations given the current state 2 Markov assumption on the states ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 78
132 References: HMMs Gales and Young (2007). The Application of Hidden Markov Models in Speech Recognition, Foundations and Trends in Signal Processing, 1 (3), : section 2.2. Jurafsky and Martin (2008). Speech and Language Processing (2nd ed.): sections ; 9.2; 9.4. (Errata at SLP2-PIEV-Errata.html) Rabiner and Juang (1989). An introduction to hidden Markov models, IEEE ASSP Magazine, 3 (1), Renals and Hain (2010). Speech Recognition, Computational Linguistics and Natural Language Processing Handbook, Clark, Fox and Lappin (eds.), Blackwells. ASR Lectures 4&5 Hidden Markov Models and Gaussian Mixture Models 79
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