1. A Tutorial on Bayesian Speech Feature Enhancement
SCALE Workshop, January 2010
A Tutorial on Bayesian Speech Feature Enhancement
Friedrich Faubel

2. I
Motivation

3. Speech Recognition System Overview
A speech recognition system converts speech to text. It basically consists of two components:
Front End: extracts speech features from the audio signal
Decoder: finds that sentence (sequence of acoustical states), which is the most likely explanation for the observed sequence of speech features
Front End
Decoder
Text
Speech

4. Speech Feature Extraction Windowing

5. Speech Feature Extraction Windowing

6. Speech Feature Extraction Windowing

7. Speech Feature Extraction Windowing

8. Speech Feature Extraction Time Frequency Analysis
Performing spectral analysis separately for each frame yields a time-frequency representation

9. Speech Feature Extraction Time Frequency Analysis
Performing spectral analysis separately for each frame yields a time-frequency representation

10. Speech Feature Extraction Perceptual Representation
Emulation of the logarithmic frequency and intensity perception of the human auditory system

11. Background Noise Background noise distorts speech features
Result: features don’t match the features used during training
Consequence: severely degraded recognition performance

12. Overview of the Tutorial
I - Motivation
II - The effect of noise to speech features
III - Transforming probabilities
IV - The MMSE solution to speech feature enhancement
V - Model-based speech feature enhancement
VI - Experimental results
VII - Extensions

13. II
Interaction Function
The Effect of Noise

14. Interaction Function + =
Principle of Superposition: signals are additive
noise
clean speech
noisy speech + =

15. Interaction Function
In the signal domain we have the following relationship:
noisy speech
noise
clean speech

16. Interaction Function
In the signal domain we have the following relationship:

17. Interaction Function
In the signal domain we have the following relationship:
After Fourier transformation, this becomes:

18. Interaction Function
In the signal domain we have the following relationship:
After Fourier transformation, this becomes:
Taking the magnitude square on both sides, we get:

19. Interaction Function
In the signal domain we have the following relationship:
After Fourier transformation, this becomes:
Taking the magnitude square on both sides, we get:

20. Interaction Function
In the signal domain we have the following relationship:
After Fourier transformation, this becomes:
Taking the magnitude square on both sides, we get:

21. Interaction Function
In the signal domain we have the following relationship:
After Fourier transformation, this becomes:
Taking the magnitude square on both sides, we get:

22. Interaction Function
In the signal domain we have the following relationship:
After Fourier transformation, this becomes:
Taking the magnitude square on both sides, we get:

23. Interaction Function
In the signal domain we have the following relationship:
After Fourier transformation, this becomes:
Taking the magnitude square on both sides, we get:

24. Interaction Function
In the signal domain we have the following relationship:
After Fourier transformation, this becomes:
Taking the magnitude square on both sides, we get:

25. Interaction Function
In the signal domain we have the following relationship:
After Fourier transformation, this becomes:
Taking the magnitude square on both sides, we get:

26. Interaction Function
In the signal domain we have the following relationship:
After Fourier transformation, this becomes:
Taking the magnitude square on both sides, we get:

27. Interaction Function
In the signal domain we have the following relationship:
After Fourier transformation, this becomes:
Taking the magnitude square on both sides, we get:

28. Interaction Function
In the signal domain we have the following relationship:
After Fourier transformation, this becomes:
Taking the magnitude square on both sides, we get:

29. Interaction Function
In the signal domain we have the following relationship:
After Fourier transformation, this becomes:
Taking the magnitude square on both sides, we get:

30. Interaction Function
In the signal domain we have the following relationship:
After Fourier transformation, this becomes:
Taking the magnitude square on both sides, we get:

31. Interaction Function
In the signal domain we have the following relationship:
After Fourier transformation, this becomes:
Taking the magnitude square on both sides, we get:

32. Interaction Function
In the signal domain we have the following relationship:
After Fourier transformation, this becomes:
Taking the magnitude square on both sides, we get:

33. Interaction Function
Taking the magnitude square on both sides, we get:

34. Interaction Function
Taking the magnitude square on both sides, we get:
Hence, in the power spectral domain we have:

35. Interaction Function
Taking the magnitude square on both sides, we get:
Hence, in the power spectral domain we have:
phase term

36. Interaction Function
Taking the magnitude square on both sides, we get:
Hence, in the power spectral domain we have:
relative phase

37. Interaction Function
The relative phase between two waves describes their relative offset in time (delay)
time
relative phase

38. Interaction Function = = = =
When 2 sound sources are present the following can happen: = =
amplification
amplification = =
attenuation
cancellation

39. Interaction Function
Taking the magnitude square on both sides, we get:
Hence, in the power spectral domain we have:
relative phase

40. Interaction Function
Taking the magnitude square on both sides, we get:
Hence, in the power spectral domain we have:
zero in average

41. Interaction Function
Taking the magnitude square on both sides, we get:
Hence, in the power spectral domain we have:
In the log power spectral domain that becomes:

42. Interaction Function
Taking the magnitude square on both sides, we get:
Hence, in the power spectral domain we have:
In the log power spectral domain that becomes:

43. Interaction Function
Taking the magnitude square on both sides, we get:
Hence, in the power spectral domain we have:
In the log power spectral domain that becomes:

44. Interaction Function
Taking the magnitude square on both sides, we get:
Hence, in the power spectral domain we have:
In the log power spectral domain that becomes:

45. Interaction Function
Taking the magnitude square on both sides, we get:
Hence, in the power spectral domain we have:
In the log power spectral domain that becomes:
Acero,
1990

46. Interaction Function
Taking the magnitude square on both sides, we get:
Hence, in the power spectral domain we have:
In the log power spectral domain that becomes:
But is that really right?

47. Interaction Function
The mean of a nonlinearly transformed random variable is not necessarily equal to the nonlinear transform of the random variable’s mean.
nonlinear
transform

48. Interaction Function
The mean of a nonlinearly transformed random variable is not necessarily equal to the nonlinear transform of the random variable’s mean.
nonlinear
transform

49. Interaction Function
Phase-averaged relationship between clean and noisy speech:

50. III
Transforming Probabilities

51. Transforming Probabilities Motivation
In the signal domain we have the following relationship:
In the log Mel domain that translates to:
nonlinear interaction function

52. Transforming Probabilities Motivation
noise power
noisy speech power
clean speech power

53. Transforming Probabilities Motivation
noisy speech power
clean speech power
noise power

54. Transforming Probabilities Motivation
clean speech power
noise power
noisy speech power

55. Transforming Probabilities Motivation

56. Transforming Probabilities Motivation
Transformation results in a non-Gaussian probability distribution for noisy speech features.

57. Transforming Probabilities Introduction
Transformation of a random variable
Transformation
Probability density function

58. Transforming Probabilities Introduction
Transformation of a random variable
Transformation
Probability density function
The transformation maps each x to a y:

59. Transforming Probabilities Introduction
Transformation of a random variable
Transformation
Probability density function
The transformation maps each x to a y:
Conversely, each y can be identified with

60. Transforming Probabilities Introduction
Transformation of a random variable
Transformation
Probability density function
Idea: use to map distribution of y to distribution of x

61. Transforming Probabilities Introduction
Transformation of a random variable
Transformation
Probability density function
Idea: use to map distribution of y to distribution of x
change of variables

62. Transforming Probabilities Introduction
Transformation of a random variable
Transformation
Probability density function
Idea: use to map distribution of y to distribution of x
Jacobian determinant

63. Transforming Probabilities Introduction
Transformation of a random variable
Transformation
Probability density function
Idea: use to map distribution of y to distribution of x
Fundamental Transformation Law of Probability

64. Transforming Probabilities Monte Carlo
Idea: approximate probability distribution by samples drawn from the distribution.
discrete probability mass
pdf

65. Transforming Probabilities Monte Carlo
Idea: approximate probability distribution by samples drawn from the distribution.
pdf
cumulative density function

66. Transforming Probabilities Monte Carlo
Idea: approximate probability distribution by samples drawn from the distribution.
Then: transform each sample
pdf
transformed pdf

67. Transforming Probabilities Monte Carlo
Idea: approximate probability distribution by samples drawn from the distribution.
Then: transform each sample
histogram
transformed pdf

68. Transforming Probabilities Local Linearization
Idea: Locally linearize the interaction function around the mean of speech and noise, using a first order Taylor series expansion.
Note: a linear transformation of a Gaussian random variable results in a Gaussian random variable.

69. Transforming Probabilities Local Linearization
Idea: Locally linearize the interaction function around the mean of speech and noise, using a first order Taylor series expansion.
Moreno,
1996
Vector Taylor Series Approach
Note: a linear transformation of a Gaussian random variable results in a Gaussian random variable.

70. Transforming Probabilities Local Linearization
Idea: Locally linearize the interaction function around the mean of speech and noise, using a first order Taylor series expansion.

71. Transforming Probabilities Local Linearization
Idea: Locally linearize the interaction function around the mean of speech and noise, using a first order Taylor series expansion.

72. Transforming Probabilities Local Linearization
Idea: Locally linearize the interaction function around the mean of speech and noise, using a first order Taylor series expansion.

73. Transforming Probabilities The Unscented Transform
Idea: similar as in Monte Carlo, select points in a determi nistic fashion and in such a way that they capture the mean and covariance of the distribution
select points

74. Transforming Probabilities The Unscented Transform
select points

75. Transforming Probabilities The Unscented Transform
select points
transform points

76. Transforming Probabilities The Unscented Transform
select points
transform points
Re-estimate parameters of the Gaussian distribution

77. Transforming Probabilities The Unscented Transform
Comparison to local linearization:
local linearization
unscented transform

78. Transforming Probabilities The Unscented Transform
select points
transform points
Re-estimate parameters of the Gaussian distribution

79. Transforming Probabilities The Unscented Transform
transform points

80. Transforming Probabilities The Unscented Transform
The points selected by the un-scented transform lie on lines around the center point.
transform points

81. Transforming Probabilities The Unscented Transform
The points selected by the un-scented transform lie on lines around the center point.
transform points

82. Transforming Probabilities The Unscented Transform
The points selected by the un-scented transform lie on lines around the center point.
transform points

83. Transforming Probabilities The Unscented Transform
The points selected by the un-scented transform lie on lines around the center point.
After nonlinear transformation, the points might no longer lie on a line
transform points

84. Transforming Probabilities The Unscented Transform
The points selected by the un-scented transform lie on lines around the center point.
After nonlinear transformation, the points might no longer lie on a line
transform points

85. Transforming Probabilities The Unscented Transform
The points selected by the un-scented transform lie on lines around the center point.
After nonlinear transformation, the points might no longer lie on a line
transform points
Hence we can measure the degree of nonlinearity as the average distance of each three points from a linear fit of the three points.

86. Transforming Probabilities The Unscented Transform
transform points
Hence we can measure the degree of nonlinearity as the average distance of each three points from a linear fit of the three points.

87. Transforming Probabilities The Unscented Transform
Hence we can measure the degree of nonlinearity as the average distance of each three points from a linear fit of the three points.
transform points

88. Transforming Probabilities The Unscented Transform
Hence we can measure the degree of nonlinearity as the average distance of each three points from a linear fit of the three points.
This can be shown to be closely related to the R2 measure used in linear regression.
transform points

89. Transforming Probabilities The Unscented Transform
true distribution
Gaussian fit
High degree of
nonlinearity
Gaussian fit does not well represent the transformed distribution

90. Transforming Probabilities An Adaptive Level of Detail Approach
Idea: splitting a Gaussian into two Gaussian components decreases the covariance and thereby the nonlinearity.

91. Transforming Probabilities An Adaptive Level of Detail Approach
Idea: splitting a Gaussian into two Gaussian components decreases the covariance and thereby the nonlinearity.
2 Gaussians

92. Transforming Probabilities An Adaptive Level of Detail Approach
Algorithm, Adaptive Level of Detail Transform [ALoDT]
start with one Gaussian g
transform that Gaussian with the UT
identify Gaussian component with highest dnl
split that component into 2 Gaussians g1, g2
transform g1 and g2 with the UT
while #(Gaussians) < N: repeat step 3.

93. Transforming Probabilities An Adaptive Level of Detail Approach
Density approximation with the Adaptive Level of Detail Transform
unscented transform

94. Transforming Probabilities An Adaptive Level of Detail Approach
Density approximation with the Adaptive Level of Detail Transform
ALoDT-2

95. Transforming Probabilities An Adaptive Level of Detail Approach
Density approximation with the Adaptive Level of Detail Transform
ALoDT-4

96. Transforming Probabilities An Adaptive Level of Detail Approach
Density approximation with the Adaptive Level of Detail Transform
ALoDT-8

97. Transforming Probabilities An Adaptive Level of Detail Approach
Density approximation with the Adaptive Level of Detail Transform
ALoDT-16

98. Transforming Probabilities An Adaptive Level of Detail Approach
Density approximation with the Adaptive Level of Detail Transform
ALoDT-32

99. Transforming Probabilities An Adaptive Level of Detail Approach
Kullback Leibler divergence between approximated and true distribution (Monte Carlo with 10M samples).
Adaptive Level of Detail Transform N 1 2 4 8 16 32
KLD
0.190
0.078
0.025
0.017
0.007
0.004
decrease by a factor of 48

100. IV
Speech Feature Enhancement
The MMSE Solution

101. Speech Feature Enhancement The MMSE Solution
Idea:
train speech recognition system on clean speech
try to map distorted features to clean speech features
Systematic Approach:
derive an estimator for clean speech given noisy speech

102. Speech Feature Enhancement The MMSE Solution
Let be an estimator for clean speech , given noisy speech .

103. Speech Feature Enhancement The MMSE Solution
Let be an estimator for clean speech , given noisy speech .
Then the expected mean square error introduced by using instead of the true is:

104. Speech Feature Enhancement The MMSE Solution
Let be an estimator for clean speech , given noisy speech .
Then the expected mean square error introduced by using instead of the true is:

105. Speech Feature Enhancement The MMSE Solution
Then the expected mean square error introduced by using instead of the true is:

106. Speech Feature Enhancement The MMSE Solution
Then the expected mean square error introduced by using instead of the true is:
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:

107. Speech Feature Enhancement The MMSE Solution
Then the expected mean square error introduced by using instead of the true is:
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:

108. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:

109. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:
But how to obtain this distribution?

110. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:
Idea: assume that the joint distribution of S and Y is Gaussian

111. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:
Idea: assume that the joint distribution of S and Y is Gaussian

112. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:
Idea: assume that the joint distribution of S and Y is Gaussian
Afify,
2007
Stereo-Based Stochastic Mapping

113. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:
Idea: assume that the joint distribution of S and Y is Gaussian

114. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:
Idea: assume that the joint distribution of S and Y is Gaussian

115. Speech Feature Enhancement The MMSE Solution
Idea: assume that the joint distribution of S and Y is Gaussian

116. Speech Feature Enhancement The MMSE Solution
Idea: assume that the joint distribution of S and Y is Gaussian
Then the conditional distribution of S|Y is again Gaussian:
with conditional mean and covariance matrix

117. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:

118. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:
Under the Gaussian assumption, this integral is easily obtained:

119. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:
Under the Gaussian assumption, this integral is easily obtained:
This is exactly what you get with the vector Taylor series approach
Moreno,
1996

120. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:
Under the Gaussian assumption, this integral is easily obtained:
Problem: speech is known to be multi modal

121. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:

122. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:
Introduce the index k of the mixture component as a hidden variable.

123. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:
Then rewrite this as

124. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:

125. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:
pull the sum out of the integral

126. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:

127. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:
independent of s

128. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:
pull this out of the integral

129. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:

130. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:
Probability that clean speech originated from the kth Gaus-sian given the noisy speech spectrum y.

131. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:
Clean speech estimate of the k-th Gaussian:

132. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:
Bayes’ theorem

133. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:

134. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:

135. Speech Feature Enhancement The MMSE Solution
Minimizing the MSE with respect to yields the optimal estimator with respect to the MMSE criterion:
joint distribution

136. V
Model-Based Speech Feature Enhancement

137. Model-Based Speech Feature Enhancement
Distribution of clean speech is modeled as Gaussian Mixture

138. Model-Based Speech Feature Enhancement
Distribution of clean speech is modeled as Gaussian Mixture + +

139. Model-Based Speech Feature Enhancement
Distribution of clean speech is modeled as Gaussian Mixture
Noise is modeled as a single Gaussian

140. Model-Based Speech Feature Enhancement
Distribution of clean speech is modeled as Gaussian Mixture
Noise is modeled as a single Gaussian

141. Model-Based Speech Feature Enhancement
Distribution of clean speech is modeled as Gaussian Mixture
Noise is modeled as a single Gaussian
Presence of noise changes the clean speech distribution according to the interaction function

142. Model-Based Speech Feature Enhancement
Distribution of clean speech is modeled as Gaussian Mixture
Noise is modeled as a single Gaussian
Presence of noise changes the clean speech distribution according to the interaction function
Construct the joint distribution of clean and noisy speech based on this model

143. Model-Based Speech Feature Enhancement
Distribution of clean speech is modeled as Gaussian Mixture
Noise is modeled as a single Gaussian
Presence of noise changes the clean speech distribution according to the interaction function
Construct the joint distribution of clean and noisy speech based on this model

144. Model-Based Speech Feature Enhancement
Construct the joint distribution of clean and noisy speech based on this model

145. Model-Based Speech Feature Enhancement
Construct the joint distribution of clean and noisy speech based on this model

146. Model-Based Speech Feature Enhancement
Construct the joint distribution of clean and noisy speech based on this model

147. Model-Based Speech Feature Enhancement
Construct the joint distribution of clean and noisy speech based on this model

148. Model-Based Speech Feature Enhancement
Noise Estimation:
Find that noise distribution, which is the most likely explanation for the observed, noisy speech features

149. Model-Based Speech Feature Enhancement
Noise Estimation:
Find that noise distribution, which is the most likely explanation for the observed, noisy speech features
mean and covariance of the noise

150. Model-Based Speech Feature Enhancement
Noise Estimation:
Find that noise distribution, which is the most likely explanation for the observed, noisy speech features
Problem: the observations are also dependent on speech!

151. Model-Based Speech Feature Enhancement
Problem: the observations are also dependent on speech!
hidden variable

152. Model-Based Speech Feature Enhancement
Problem: the observations are also dependent on speech!
hidden variable

153. Model-Based Speech Feature Enhancement
Problem: the observations are also dependent on speech!
hidden variable

154. Model-Based Speech Feature Enhancement
Noise Estimation:
Find that noise distribution, which is the most likely explanation for the observed, noisy speech features
Problem: the observations are also dependent on speech!

155. Model-Based Speech Feature Enhancement
Noise Estimation:
Find that noise distribution, which is the most likely explanation for the observed, noisy speech features
Problem: the observations are also dependent on speech!
Hence, the Expectation Maximization algorithm is used.
Rose,
1994
Moreno,
1996

156. Model-Based Speech Feature Enhancement
Expectation Step: construct the joint distribution by using the current noise parameter estimate Then calculate

157. Model-Based Speech Feature Enhancement
Expectation Step: construct the joint distribution by using the current noise parameter estimate Then calculate

158. Model-Based Speech Feature Enhancement
Expectation Step: construct the joint distribution by using the current noise parameter estimate Then calculate
Maximization Step: Reestimate by ac-cumulating statistics of the instantaneous noise estimates for each possible , weighted by the probability that clean speech originated from this Gaussian.

159. Model-Based Speech Feature Enhancement
Maximization Step: Reestimate by ac-cumulating statistics of the instantaneous noise estimates for each possible , weighted by the probability that clean speech originated from this Gaussian.

160. Model-Based Speech Feature Enhancement
Maximization Step: Reestimate by ac-cumulating statistics of the instantaneous noise estimates for each possible , weighted by the probability that clean speech originated from this Gaussian:

161. Model-Based Speech Feature Enhancement
Maximization Step: Reestimate by ac-cumulating statistics of the instantaneous noise estimates for each possible , weighted by the probability that clean speech originated from this Gaussian:

162. Model-Based Speech Feature Enhancement
Maximization Step: Reestimate by ac-cumulating statistics of the instantaneous noise estimates for each possible , weighted by the probability that clean speech originated from this Gaussian:
But how to obtain this distribution?

163. Model-Based Speech Feature Enhancement
Maximization Step: Reestimate by ac-cumulating statistics of the instantaneous noise estimates for each possible , weighted by the probability that clean speech originated from this Gaussian:

164. Model-Based Speech Feature Enhancement
Maximization Step: Reestimate by ac-cumulating statistics of the instantaneous noise estimates for each possible , weighted by the probability that clean speech originated from this Gaussian:

165. Model-Based Speech Feature Enhancement
Maximization Step: Reestimate by ac-cumulating statistics of the instantaneous noise estimates for each possible , weighted by the probability that clean speech originated from this Gaussian:

166. Model-Based Speech Feature Enhancement
Maximization Step: Reestimate by ac-cumulating statistics of the instantaneous noise estimates for each possible , weighted by the probability that clean speech originated from this Gaussian:

167. Model-Based Speech Feature Enhancement
Maximization Step: Reestimate by ac-cumulating statistics of the instantaneous noise estimates for each possible , weighted by the probability that clean speech originated from this Gaussian:
So, we have , need

168. Model-Based Speech Feature Enhancement
Maximization Step: Reestimate by ac-cumulating statistics of the instantaneous noise estimates for each possible , weighted by the probability that clean speech originated from this Gaussian:
But that is just the conditional Gaussian distribution
with conditional mean
and covariance

169. Model-Based Speech Feature Enhancement
Maximization Step: Reestimate by ac-cumulating statistics of the instantaneous noise estimates for each possible , weighted by the probability that clean speech originated from this Gaussian:

170. VI
Experimental Results

171. Experimental Results Speech Recognition Experiments
clean speech from MC-WSJ-AV corpus
noise from the NOISEX-92 database (artifically added)
MFCC with 13 components, stacking of 15 frames, LDA
cepstral mean and variance normalization
1743 acoustical states; Gaussians

172. Experimental Results
WER, destroyer engine noise

173. Experimental Results
WER, factory noise

174. VII
Extensions

175. Extensions Sequential noise estimation:
Sequential expectation maximization (SEM), Kim, 1998

176. Extensions Sequential noise estimation:
Sequential expectation maximization (SEM), Kim, 1998
Interacting Multiple Model (IMM) Kalman Filter, Kim, 1999

177. Extensions Sequential noise estimation:
Sequential expectation maximization (SEM), Kim, 1998
Interacting Multiple Model (IMM) Kalman Filter, Kim, 1999
Particle filter, Yao, 2001

178. Extensions Sequential noise estimation:
Sequential expectation maximization (SEM), Kim, 1998
Interacting Multiple Model (IMM) Kalman Filter, Kim, 1999
Particle filter, Yao, 2001
Improve speech recognition through:
Combination with Joint Uncertainty Decoding, Shinohara, 2008

179. Extensions Sequential noise estimation:
Sequential expectation maximization (SEM), Kim, 1998
Interacting Multiple Model (IMM) Kalman Filter, Kim, 1999
Particle filter, Yao, 2001
Improve speech recognition through:
Combination with Joint Uncertainty Decoding, Shinohara, 2008
Combination with bounded conditional mean imputation?