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1 Aalborg Universitet Model-based Noise PSD Estimation from Speech in Non-stationary Noise Nielsen, Jesper jær; avalekalam, Mathew Shaji; Christensen, Mads Græsbøll; Boldt, Jesper Bünsow Published in: IEEE International Conference on Acoustics, Speech, and Signal Processing Creative Commons License Unspecified Publication date: 2018 Document Version Other version Link to publication from Aalborg University Citation for published version (APA): Nielsen, J.., avalekalam, M. S., Christensen, M. G., & Boldt, J. B. (2018). Model-based Noise PSD Estimation from Speech in Non-stationary Noise. In IEEE International Conference on Acoustics, Speech, and Signal Processing Calgary, Canada: IEEE. I E E E International Conference on Acoustics, Speech and Signal Processing. Proceedings General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.? Users may download and print one copy of any publication from the public portal for the purpose of private study or research.? You may not further distribute the material or use it for any profit-making activity or commercial gain? You may freely distribute the URL identifying the publication in the public portal? Take down policy If you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from vbn.aau.dk on: december 07, 2018

2 Variational Bayesian Inference for Model-based Noise PSD Estimation Jesper jær Nielsen 1, Mathew Shaji avalekam 1, Mads Græsbøll Christensen 1, and Jesper Boldt 2 1 Audio Analysis Lab, CREATE, Aalborg University 2 GN ReSound Last compiled: April 17, Introduction This document gives the details of the variational Bayesian (VB) noise power spectral density (PSD) estimator proposed in our ICASSP-paper 2. In the first section of the document, a general description of the VB approach is given to joint parameter estimation and model comparison. If the reader is familiar with the VB approach, this section can be skipped. In the second section, the VB algorithm is applied to the model described in 2. We strongly encourage the reader to read the ICASSP paper before continuing reading this document. 2 A VB tutorial Suppose we wish to compute the posterior distribution p(θ x) = p(θ x, M k )p(m k x), (1) for the model parameters θ, but cannot do this analytically. Sometimes, we can simplify the problem considerably if we introduce latent variables s. An example of this is for Gaussian mixture models where the introduction of latent variables allows us to use the EM-algorithm to produce a solution. If we let z = s T and the model parameters is p(z x) = θ T T, the joint posterior over the latent parameters p(z x, M k )p(m k x). (2) 1

3 Often, this posterior cannot be computed analytically, so we assume that it factorises for each model M k as p(z x, M k ) q k (z) = M i=1 q ik (z i ) (3) where we have divided z into M non-overlapping subgroups. In the sequel, we will use the more compact notation q ik = q ik (z i ). Also note that even though we do not assume any factorisation for p(m k x), we will only obtain an approximation q(m k ) of it since we use q k (z) instead of p(z x, M k ). Thus, we have that p(z, M k x) q(z, M k ) = q(m k )q k = q(m k ) M i=1 q ik. (4) Thus, we have now reformulated the problem from finding p(z, M k x) to that of finding the different factors q ik and q(m k ). If we select the factorisation in a clever way, it is much easier to find these factors (using VB) than it is to find the exact posterior p(z, M k x). The art of applying the VB framework is, therefore, to elicit a signal model (i.e., an observation model p(x z, M k ) and prior distributions p(z M k ) and p(m k )) and a factorisation so that the various factors can easily be computed while still being a good approximation to the exact posterior. In the sequel, we assume that we have elicited these and focus on describing how the various factors of the elicited factorisation are computed from the signal model. For any pdf q, we have that the log of the model evidence can be written as 1, p. 473 ln p(x) = L(q) q(m k ) q k ln p(z, M k x) dz (5) q k q(m k ) where L(q) = q(m k ) q k ln p(z, x, M k) dz. (6) q k q(m k ) It turns out that maximising the lower bound L(q) w.r.t. q produces the best approximation to the posterior p(z, M k x). To perform this maximisation, the lower bound is first rewritten as L(q) = q(m k )L k (q k ) + q(m k ) ln p(m k) q(m k ) (7) 2

4 where L k (q k ) = = = = = q k ln p(z, x M k) dz q k (8) q k ln p(z, x M k )dz q k ln q k dz (9) ( ) M ( ) q ik dz i dz j ln q ik q ik dz i i=1 i=1 q jk ln p(z, x M k ) (10) ( ) M q jk ln p(z, x M k ) q ik dz i dz j q ik ln q ik dz i (11) i=1 ( ) q jk ln p(z, x M k ) q ik dz i ln q jk dz j q ik ln q ik dz i. (12) Given the data x and the pdfs q ik for i = j, the integral in the bracket is a function of z j. However, this function does not necessarily integrate to one, but such a function can easily be defined as ln p(z j x, M k ) = E ln p(z, x M k ) ln Z jk (13) where E ln p(z, x M k ) and Z jk are the expectation operator w.r.t. to q ik and the normalisation constant so that p(z j x, M k ) integrates to one, respectively. These are given by ( ) E ln p(z, x M k ) = ln p(z, x M k ) q ik dz i (14) Z jk = exp { E ln p(z, x M k ) } dz j. (15) Consequently, we can now write L k (q k ) as L k (q k ) = q jk ln p(zj x, M k ) + ln Z jk ln q jk dzj q ik ln q ik dz i (16) = L(q jk p(z j x, M k )) + ln Z jk + Hq ik (17) where L(q jk p(z j x, M k )) and Hq ik are the ullback-leibler (L) divergence and the entropy, respectively. We have now written the lower bound L(q) in a form which allows us to maximise it w.r.t. to q jk. Since the L divergence is the only term which depends on the functional q jk, we maximise the lower bound by minimising the L divergence. Fortunately, this is easy since the minimum of the L divergence is zero, and this minimum value is obtained if and only if q jk = p(z j x, M k ) (18) 3

5 and the solution to the optimisation problem, therefore, is that ln q jk = E ln p(z, x M k ) + const. (19) The above solution is not a closed-form solution since the expectation operator depends on {q ik } which are also unknown. However, if we optimise for the individual q jk s iteratively, we are guaranteed to converge to a solution 1, p The VB Lower Bound For every model order, we can keep an eye on the convergence by monitoring the lower bound L k (q k ). For the optimal form of q jk, the L divergence vanishes, and the lower bound is given by L k (q k ) = ln Z jk + Hq ik. (20) Since this should hold for all j s, we can stop the algorithm when L k (q k ) is nearly the same for all j s. An alternative formulation for the lower bound can also be constructed directly from the definition of L k (q k ) as L k (q k ) = q k ln p(z, x M k )dz q k ln q k dz (21) = 2.2 The VB Model Comparison q k ln p(x z, M k )dz + q k ln p(z M k )dz + = E qk ln p(x z, M k ) + E qk ln p(z M k ) + M i=1 M i=1 Hq ik (22) Hq ik. (23) We can also do model comparison in the VB framework. To do this, we rewrite L(q) as L(q) = = q(m k )L k (q k ) + q(m k ) ln p(m k) q(m k ) (24) q(m k ) ln (p(m k ) exp(l k (q k ))) ln q(m k ). (25) Given the data x and the joint pdf q k, the first term in the bracket is a function of the model M k. This function, however, does not necessarily integrate to one, but such a function can easily be defined as ln p(m k x) = L k (q k ) + ln p(m k ) ln Z k (26) where Z k is a normalisation constant given by Z k = p(m k ) exp L k (q k ). (27) 4

6 Using these definitions, we can write L(q) as L(q) = L(q(M k ) p(m k x)) + ln Z k (28) which is clearly maximised for or, equivalently, q(m k ) = p(m k x) (29) ln q(m k ) = L k (q k ) + ln p(m k ) + const. (30) = ln Z jk + Hq ik + ln p(m k ) + const. (31) for any j. 3 Noise PSD Estimation Using VB We will now apply the VB framework to solve the problem of computing the posterior distribution on the excitaiton noise variances from a mixture of two periodic AR processes. That is, we observe y = s + e (32) where p(e σ 2 e, M k ) = N (0, σ 2 e R e (b k )) (33) p(s σ 2 s, M k ) = N (0, σ 2 s R s (a k )) (34) R s (a k ) = N 1 FD s (a k )F H (35) R e (b k ) = N 1 FD e (b k )F H (36) F nl = exp(j2π(n 1)(l 1)/N), n, l = 1,..., N (37) Λ s (a k ) = diag(f H a T k 0 T ) (38) T) Λ e (b k ) = diag(f H bk 0 T (39) ( 1 D s (a k ) = Λs H (a k )Λ s (a k )) (40) ( 1 D e (b k ) = Λe H (b k )Λ e (b k )). (41) Moreover, we assume that the prior for the excitation noise variances are given by p(σ 2 e M k ) = Inv-G(α e,k, β e,k ) (42) p(σ 2 s M k ) = Inv-G(α s,k, β s,k ) (43) 5

7 whose hyperparameters we initially assume known. The latens variable of our model is s whereas the model parameters are σ 2 s and σ 2 e. Note that the AR parameters are given by the model. From the above model, it also follows that p(y s, σ 2 e, M k ) = N (s, σ 2 e R e (b k )) (44) so that joint distribution over the observations, the latent variables, and the model parameters factorise as p(y, s, σ 2 s, σ 2 e M k ) = p(y s, σ 2 e, M k )p(s σ 2 s, M k )p(σ 2 s M k )p(σ 2 e M k ). (45) In our VB algorithm, we seek to compute and approximation of the joint posterior p(s, σ 2 s, σ 2 e M k ) using the factorisation p(s, σ 2 s, σ 2 e M k ) q k (s, σ 2 s, σ 2 e ) = q 1k (s)q 2k (σ 2 s )q 3k (σ 2 e ) (46) where the factors can be viewed as approximations to the marginal posteriors with q 1k (s) p(s y, M k ) (47) q 2k (σ 2 s ) p(σ 2 s y, M k ) (48) q 3k (σ 2 e ) p(σ 2 e y, M k ). (49) We also wish to compute an approximation q(m k ) of the posterior pmf for the models. Before we can do that, however, we first have to find the functional expression of the factors q ik, their entropies Hq ik, and the normalisation constants Z ik. 3.1 Functional expressions for the factor q 1k (s) According to the above tutorial, we have that ln q 1k (s) = E i =1 ln p(z, y M k ) + const. (50) Inserting the expression for p(z, y M k ) in the expectation operator, we obtain E i =1 ln p(z, y M k ) = E i =1 ln p(y s, σ 2 e, M k ) + E i =1 ln p(s σ 2 s, M k ) + E i =1 ln p(σ 2 s M k ) + E i =1 ln p(σ 2 e M k ). (51) Of the four terms, the first two terms depend on s whereas the last two do not. However, we still have to compute the last two terms in order to compute the normalisation constant Z 1k. 6

8 Since ln p(y s, σ 2 e, M k ) = (N/2) ln(2πσ 2 e ) (1/2) ln det(r e (b k )) 1 2σe 2 (y s) T Re 1 (b k )(y s) (52) ln p(s σs 2, M k ) = (N/2) ln(2πσs 2 ) (1/2) ln det(r s (a k )) 1 2σs 2 s T Rs 1 (a k )s (53) ln p(σ 2 s M k ) = α s,k ln β s,k ln Γ(α s,k ) (α s,k + 1) ln σ 2 s β s,k σ 2 s ln p(σ 2 e M k ) = α e,k ln β e,k ln Γ(α e,k ) (α e,k + 1) ln σ 2 e β e,k σ 2 e (54), (55) we have for the four terms that E i =1 ln p(y s, σe 2, M k ) = q 3k (σe 2 ) ln p(y s, σe 2, M k )dσe 2 (56) = (N/2) ln(2π) (N/2) E q3k ln σ 2 e (1/2) ln det(r e (b k )) (1/2)(y s) T Re 1 (b k )(y s) E q3k σe 2 (57) = ln N (s, E 1 q 3k σe 2 R e (b k )) (N/2) { ln E q3k σe 2 + E q3k ln σe 2 } (58) E i =1 ln p(s σs 2, M k ) = q 2k (σs 2 ) ln p(s σs 2, M k )dσs 2 (59) = (N/2) ln(2π) (N/2) E q2k ln σ 2 s (1/2) ln det(r s (a k )) (1/2)s T Rs 1 (a k )s E q2k σs 2 (60) = ln N (0, E 1 q 2k σs 2 R s (a k )) (N/2) { ln E q2k σ 2 s + E q2k ln σ 2 s } (61) E i =1 ln p(σ 2 s M k ) = α s,k ln β s,k ln Γ(α s,k ) (α s,k + 1) E q2k ln σ 2 s β s,k E q2k σ 2 s (62) E i =1 ln p(σ 2 e M k ) = α e,k ln β e,k ln Γ(α e,k ) (α e,k + 1) E q3k ln σ 2 e β e,k E q3k σ 2 e (63) By only retaining the terms from these four expressions which depend on s, we obtain that q 1k (s) N (s, E 1 from which we can derive that q 3k σe 2 R e (b k ))N (0, E 1 q 2k σ 2 s R s (a k )) (64) q 1k (s) = N (ŝ, Σŝ) (65) 1 Σŝ = E q2k σs 2 Rs 1 (a k ) + E q3k σe 2 Re 1 (b k ) (66) ŝ = Σŝ E q3k σe 2 Re 1 (b k )y. (67) 7

9 and that N (s, E 1 q 3k σe 2 R e (b k ))N (0, E 1 q 2k σ 2 s R s (a k ))ds = N (0, E 1 q 2k σs 2 R s (a k ) + E 1 q 3k σ 2 e R e (b k )) (68) by using standard Bayesian inference. Thus, the log-normalisation factor ln Z 1k is given by ln Z 1 = ln exp { E i =1ln p(z, y M k ) } ds (69) = ln N (0, E 1 q 2k σs 2 R s (a k ) + E 1 q 3k σ 2 e R e (b k )) (N/2) { ln E q2k σs 2 + E q2k ln σs 2 + ln E q3k σe 2 + E q3k ln σe 2 } (70) E i =1 ln p(σs 2 M k ) + E i =1 ln p(σe 2 M k ). (71) Finally, since q 1k (s) is a multivariate Gaussian distribution, its entropy is given by Hq 1k = (N/2) ln(2π exp(1)) + (1/2) ln det(σŝ). (72) 3.2 Functional expressions for the factor q 2k (σ 2 s ) According to the above tutorial, we have that ln q 2k (σ 2 s ) = E i =2 ln p(z, y M k ) + const. (73) Inserting the expression for p(z, y M k ) in the expectation operator, we obtain E i =2 ln p(z, y M k ) = E i =2 ln p(y s, σ 2 e, M k ) + E i =2 ln p(s σ 2 s, M k ) + E i =2 ln p(σ 2 s M k ) + E i =2 ln p(σ 2 e M k ). (74) Of the four terms, the second and third terms depend on σs 2 whereas the first and fourth do not. However, we still have to compute all the terms to obtain the normalisation constant Z 2k. For the four terms, we have that E i =2 ln p(y s, σe 2, M k ) = q 1k (s)q 3k (σe 2 ) ln p(y s, σe 2, M k )dsdσe 2 (75) = (N/2) ln(2π) (N/2) E q3k ln σe 2 (1/2) ln det(r e (b k )) (1/2) E q1k (y s) T Re 1 (b k )(y s) E q3k σe 2 (76) E i =2 ln p(s σs 2, M k ) = q 1k (s) ln p(s σs 2, M k )ds (77) = (N/2) ln(2π) (N/2) ln σs 2 (1/2) ln det(r s (a k )) 1 2σs 2 E q1k s T Rs 1 (a k )s E i =2 ln p(σs 2 M k ) = α s,k ln β s,k ln Γ(α s,k ) (α s,k + 1) ln σs 2 β s,k σs 2 (79) E i =2 ln p(σe 2 M k ) = E i =1 ln p(σe 2 M k ) (80) (78) 8

10 From this, we have that ln q 2k (σs 2 ) = (N/2 + α s,k + 1) ln σs 2 (1/2) E q 1k s T Rs 1 (a k )s + β s,k σs 2 + const. (81) where all the terms independent of σ 2 s are absorbed in the constant. Thus, where q 2k (σ 2 s ) = Inv-G(a s,k, b s,k ) (82) a s,k = N/2 + α s,k (83) b s,k = (1/2) E q1k s T Rs 1 (a k )s + β s,k. (84) The log-normalisation factor ln Z 2 can now also be found to ln Z 2 = ln = ln exp { E i =2ln p(z, y M k ) } dσs 2 ( (σs 2 ) (as,k+1) exp ) dσ 2 s + α s,k ln β s,k ln Γ(α s,k ) b s,k σs 2 (N/2) ln(2π) (N/2) ln σs 2 (1/2) ln det(r s (a k )) (85) + E i =2 ln p(y s, σ 2 e, M k ) + E i =2 ln p(σ 2 e M k ) (86) = a s,k ln b s,k + ln Γ(a s,k ) + α s,k ln β s,k ln Γ(α s,k ) (N/2) ln(2π) (N/2) ln σ 2 s (1/2) ln det(r s (a k )) + E i =2 ln p(y s, σ 2 e, M k ) + E i =2 ln p(σ 2 e M k ) (87) Finally, since q 2k (σ 2 s ) is an inverse Gamma distribution, its entropy is given by where Ψ( ) is the digamma function. Hq 2k = a s,k + ln(b s,k Γ(a s,k )) (1 + a s,k )Ψ(a s,k ) (88) 3.3 Functional expressions for the factor q 3k (σ 2 e ) According to the above tutorial, we have that ln q 3k (σ 2 e ) = E i =3 ln p(z, y M k ) + const. (89) Inserting the expression for p(z, y M k ) in the expectation operator, we obtain E i =3 ln p(z, y M k ) = E i =3 ln p(y s, σ 2 e, M k ) + E i =3 ln p(s σ 2 s, M k ) + E i =3 ln p(σ 2 s M k ) + E i =3 ln p(σ 2 e M k ). (90) 9

11 Of the four terms, the first and the fourth terms depend on σe 2 whereas the second and third do not. However, we still have to compute all the terms to obtain the normalisation constant Z 3k. For the four terms, we have that E i =3 ln p(y s, σe 2, M k ) = q 1k (s) ln p(y s, σe 2, M k )ds (91) = (N/2) ln(2π) (N/2) ln σe 2 (1/2) ln det(r e (b k )) 1 2σe 2 E q1k (y s) T Re 1 (b k )(y s) (92) E i =3 ln p(s σs 2, M k ) = q 1k (s)q 2k (σs 2 ) ln p(s σs 2, M k )dsdσs 2 (93) = (N/2) ln(2π) (N/2) E q2k ln σs 2 (1/2) ln det(r s (a k )) (1/2) E q1k s T Rs 1 (a k )s E q2k σs 2 (94) E i =3 ln p(σ 2 s M k ) = E i =1 ln p(σ 2 s M k ) (95) E i =3 ln p(σ 2 e M k ) = α e,k ln β e,k ln Γ(α e,k ) (α e,k + 1) ln σ 2 e β e,k σ 2 e (96) (97) From this, we have that ln q 3k (σ 2 e ) = (N/2 + α e,k + 1) ln σ 2 e (1/2) E q1k (y s) T Re 1 (b k )(y s) + β e,k σe 2 + const. (98) where all the terms independent of σ 2 e are absorbed in the constant. Thus, where q 3k (σ 2 s ) = Inv-G(a e,k, b e,k ) (99) a e,k = N/2 + α e,k (100) b e,k = (1/2) E q1k (y s) T Re 1 (b k )(y s) + β e,k. (101) 10

12 The log-normalisation factor ln Z 3 can now also be found to ln Z 3 = ln exp { E i =3ln p(z, y M k ) } dσ3 2 = ln ( (σe 2 ) (ae,k+1) exp ) dσ 2 e + α e,k ln β e,k ln Γ(α e,k ) b e,k σe 2 (N/2) ln(2π) (N/2) ln σe 2 (1/2) ln det(r e (b k )) (102) + E i =3 ln p(s σ 2 e, M k ) + E i =3 ln p(σ 2 s M k ) (103) = a e,k ln b e,k + ln Γ(a e,k ) + α e,k ln β e,k ln Γ(α e,k ) (N/2) ln(2π) (N/2) ln σ 2 e (1/2) ln det(r e (b k )) + E i =3 ln p(s σ 2 e, M k ) + E i =3 ln p(σ 2 s M k ) (104) Finally, since q 3k (σ 2 e ) is an inverse Gamma distribution, its entropy is given by 3.4 Evaluating the expectations Hq 3k = a e,k + ln(b e,k Γ(a e,k )) (1 + a e,k )Ψ(a e,k ). (105) To summarize the main result above, we have that where Σŝ = E q2k σ 2 s q 1k (s) = N (ŝ, Σŝ) (106) q 2k (σ 2 s ) = Inv-G(a s,k, b s,k ) (107) q 3k (σ 2 e ) = Inv-G(a e,k, b e,k ) (108) R 1 s 1 (a k ) + E q3k σe 2 Re 1 (b k ) (109) ŝ = Σŝ E q3k σe 2 Re 1 (b k )y (110) a s,k = N/2 + α s,k (111) b s,k = (1/2) E q1k s T Rs 1 (a k )s + β s,k (112) a e,k = N/2 + α e,k (113) b e,k = (1/2) E q1k (y s) T Re 1 (b k )(y s) + β e,k. (114) Since we now have derived the form of the distributions for the factors, we can evaluate the expectations. These are E q2k σs 2 = a s,k (115) b s,k E q3k σe 2 = a e,k (116) b e,k ( ) E q1k s T Rs 1 (a k )s = ŝ T Rs 1 (a k )ŝ + tr Rs 1 (a k )Σŝ (117) ( ) E q1k (y s) T Rs 1 (a k )(y s) = ê T Re 1 (b k )ê + tr Re 1 (b k )Σŝ (118) 11

13 where ê = y ŝ. For the evaluation of the normalisation constants, we also have that 3.5 Computing the model factor q(m k ) E q2k ln σ 2 s = ln b s,k Ψ(a s,k ) (119) E q3k ln σ 2 e = ln b e,k Ψ(a e,k ). (120) To evaluate the model factor q(m k ), we have to compute the lower bound L k (q k ) for all models. This lower bound can be computed from the normalisation factors {Z ik } 3 i=1 and the entropies for q 1k (s), q 2k (σs 2 ), and q 3k (σe 2 ). Alternatively, we also have from (23) that L k (q k ) can be written as L k (q k ) = E qk ln p(y s, σ 2 e, M k ) + E qk ln p(s σ 2 s, M k )+ E qk ln p(σ 2 s M k ) + E qk ln p(σ 2 e M k ) + Since none of the expectations depends on all three sets of unknowns, we have that M i=1 Hq ik. (121) E qk ln p(y s, σ 2 e, M k ) = E i =2 ln p(y s, σ 2 e, M k ) (122) = (N/2) ln(2π) (N/2) E q3k ln σe 2 (1/2) ln det(r e (b k )) (1/2) E q1k (y s) T Re 1 (b k )(y s) E q3k σe 2 (123) E qk ln p(s σ 2 s, M k ) = E i =3 ln p(s σ 2 s, M k ) (124) References = (N/2) ln(2π) (N/2) E q2k ln σs 2 (1/2) ln det(r s (a k )) (1/2) E q1k s T Rs 1 (a k )s E q2k σs 2 (125) E qk ln p(σ 2 s M k ) = E i =1 ln p(σ 2 s M k ) = E i =3 ln p(σ 2 s M k ) (126) = α s,k ln β s,k ln Γ(α s,k ) (α s,k + 1) E q2k ln σ 2 s β s,k E q2k σs 2 (127) E qk ln p(σe 2 M k ) = E i =1 ln p(σe 2 M k ) = E i =2 ln p(σe 2 M k ) (128) = α e,k ln β e,k ln Γ(α e,k ) (α e,k + 1) E q3k ln σ 2 e β e,k E q3k σ 2 e. (129) 1 C. M. Bishop. Pattern Recognition and Machine Learning. New York, NY, USA: Springer, isbn: J.. Nielsen et al. Model-based Noise PSD Estimation from Speech in Non-stationary Noise. In: Proc. IEEE Int. Conf. Acoust., Speech, Signal Process