2015 IEEE INTERNATIONAL WORKSHOP ON MACHINE LEARNING FOR SIGNAL PROCESSING, SEPT. 17–20, 2015, BOSTON, USA

MODELING SPEECH PARAMETER SEQUENCES WITHLATENT TRAJECTORY HIDDEN MARKOV MODEL
Hirokazu Kameoka
Nippon Telegraph and Telephone Corporation / The University of Tokyo

ABSTRACTThis paper proposes a probabilistic generative model of a sequence of vectors called the latent trajectory hidden Markov
model (HMM). While a conventional HMM is only capable ofdescribing piecewise stationary sequences of data vectors, theproposed model is capable of describing continuously timevarying sequences of data vectors, governed by discrete hidden states. This feature is noteworthy in that it can be used
to model many kinds of time series data that are continuousin nature such as speech spectra. Given a sequence of observed data, the optimal state sequence can be decoded using
the expectation-maximization (EM) algorithm. Given a set oftraining examples, the underlying model parameters can betrained by either the expectation-maximization algorithm or
the variational inference algorithm.
Index Terms— Sequential modeling, Hidden Markov
model (HMM), Trajectory HMM, Latent trajectory HMM,
Expectation-Maximization algorithm, variational inference
1. INTRODUCTION
The weakness of hidden Markov models (HMMs) is that they
have difficulty in modeling and capturing the local dynamics of feature sequences due to the piecewise stationarity assumption and the conditional independence assumption on
feature sequences. Traditionally, in speech recognition systems, this limitation has been circumvented by appending dynamic (delta and delta-delta) components to the feature vectors. HMM-based speech synthesis systems [1] also use thejoint vector of static and dynamic features as an observed
vector in the training process. In the synthesis process, on
the other hand, a sequence of static features is generated according to the output probabilities of the trained HMM given
an input sentence by taking account of the explicit constraint
between the static and dynamic features [2]. Although the
HMM-based speech synthesis framework has many attractive
features, one drawback is that the criteria used for training andsynthesis are inconsistent. While the joint likelihood of staticand dynamic features is maximized during the training process, the likelihood of only the static features is maximized
during the synthesis process. This implies that the model parameters are not trained in such a way that the generated paThis work was supported by JSPS KAKENHI Grant Numbers 26730100
and 26280060. The author would like to thank Dr. Tomoki Toda (NAIST)
for fruitful discussions and Mr. Tomohiko Nakamura (University of Tokyo)
for his help with conducting the experiments.

c
978-1-4673-7454-5/15/$31.00 ⃝2015
IEEE

rameter sequences become optimal. To address this problem,
Zen [3] introduced a variant of HMM called the “trajectory
HMM,” which was obtained by incorporating the explicit relationship between static and dynamic features into the traditional HMM. This has made it possible to provide a unified
framework for the training and synthesis of speech parametersequences, however, it causes difficulty as regards parameter
inference. Since the conditional independence assumption on
the feature vectors is lost, efficient algorithms for training anddecoding regular HMMs such as the Viterbi algorithm and the
Forward-Backward algorithm are no longer applicable to the
trajectory HMM. Thus, some approximations and brute-force
methods are usually necessary to obtain training and decoding
algorithms [3, 4].
In this paper, we propose formulating a new model called
the “latent trajectory HMM.” In contrast with the conventional trajectory HMM, the present model splits the generative process of an observed feature sequence into two processes, one for a sequence of the joint vectors of static anddynamic features given HMM states and the other for an observed feature sequence given the sequence of the joint vectors. By treating the joint vector of static and dynamic features as a latent variable to be marginalized out, we obtain aprobability density function of an observed feature sequencewith a different form from the likelihood function of the trajectory HMM. As described below, this new formulation naturally allows the combined use of powerful inference techniques such as the expectation-maximization (EM) algorithm,
Viterbi algorithm and Forward-Backward algorithm for training and decoding, while still retaining the spirit of the original
trajectory HMM.
This work is not only directed towards speech synthesis
applications but also towards several different applications
such as voice conversion and acoustic-to-articulatory mapping, in which trajectory modeling has proven to be effective [5–7]. Another interesting application we have in mind
is audio source separation. Recently, we proposed methods
for single- and multi-channel audio source separation basedon factorial HMMs [8–11], where the spectrogram of a mixture signal is modeled as the sum of the outputs emitted frommultiple HMMs, each representing the spectrogram of an underlying source. One promising way to improve this approach
would be to incorporate the dynamics of source spectra. Thiscan be accomplished by plugging the present model into thefactorial HMM formulation. The present formulation willplay a key role in making this possible.

2. TRAJECTORY HIDDEN MARKOV MODEL
We start by briefly reviewing the original formulation of
the trajectory HMM [3]. Let us use ct to denote a Ddimensional static feature vector and define the joint vectorof ct and its velocity and acceleration components ot :=
T
2 T T
3D
[cT
as the observed vector at time t.
t , ∆ct , ∆ ct ] ∈ R
We write the sequences of the static features and the observed
T T
T
T T
vectors as c = [cT
1 , . . . , cT ] and o = [o1 , . . . , oT ] , respectively. Thus, the dimensions of c and o become DT and
3DT . The relationship between c and o can be described
explicitly using a constant 3DT by DT matrix W aso = W c,(1)where W is a sparse matrix that appends first and second
order time derivatives to the static feature vector sequence.
Within the traditional HMM framework, a sequence ofobserved vectors, o, is simply assumed to be generated froman HMM. Here, if we assume the emission probability density
to be a single Gaussian distribution, the probability density
function of o given a state sequence s = [s1 , . . . , sT ] and an
HMM parameter set λ = {µ, U , π}, with µ = {µi }1≤i≤I ,U = {U i }1≤i≤I , and π = {π i,j }1≤i≤I,1≤j≤J , is given asp(o|s, λ) = N (o; µs , U s ) =T∏N (ot ; µst , U st ),

(2)

t=1

where N (x; µ, Σ) denotes a Gaussian distribution with mean
µ and covariance Σ:N (x; µ, Σ) ∝1
|Σ|e− 2 (x−µ) Σ
1

1
2

T

−1

(x−µ)

.

(3)

µs and U s denote the mean sequence and a block diagonal
matrix whose diagonal elements are given by the sequence ofthe covariance matrices of the emission densities over time:T Tµs = [µTs1 , . . . , µsT ] ,(4)

U s = diag(U s1 , . . . , U sT ).

(5)

In HMM-based speech synthesis systems, the parameter set istypically trained by solving the maximum likelihood estimation problem∑
λ̂ = argmax logp(o|s, λ)p(s),
(6)
λ

s

where p(s) is given by the product of the state transition probabilities. At the synthesis stage, given a state sequence s andwith the trained parameter λ, a static feature sequence c isgenerated according toĉ = argmax p(c|s, λ),(7)cwhere p(c|s, λ) is defined asp(c|s, λ) ∝ N (W c; µs , U s )
∝e

T
T
−1
− 12 (cT W T U −1
s W c−2c W U s µs )

(8)
(9)

Fig. 1. Illustrations that show the difference between HMM
and trajectory HMM.
= N (c; c̄s , V s ).

(10)

By completing the square in the exponent of (9) with respect
to c, we immediately obtain c̄s and V s as
−1c̄s = (W T U −1
W T U −1s W)
s µs ,

(11)

−1
U −1
.
s W)

(12)

V s = (WT

Thus, the solution to (7) is c̄s . Geometrically, (10) can beviewed as a cutting plane of the density p(o|s, λ) at o = W c.
As shown above, the traditional HMM-based framework uses different criteria for training and synthesis: Whilep(o|s, λ) is used for training, p(c|s, λ) is used for synthesis.
This implies that λ̂ is not necessarily optimal for generating
optimal c. To address this inconsistency between the training
and synthesis criteria, Zen [3] proposed introducing a framework called the “trajectory HMM”, which also uses (10) as
the training criterion. Instead of solving (6), the parameterset λ is thus trained by solving{λ̂, ŝ} = argmax log p(c|s, λ)p(s),

(13)

λ,s

where c is treated as the observed data.
Unlike the regular HMM, the conditional independence
assumption on observed vectors does not hold in the trajectory HMM: While the regular HMM assumes that each observed vector depends only on the current state, (10) indicates that the observed vector ct at each frame depends onthe entire state sequence. This implies that it is difficult to
directly apply the efficient decoding and training algorithms
used in the HMM framework (such as the Viterbi algorithm
and the Forward-Backward algorithm). Thus, some approximations and brute-force methods are usually necessary to perform training and decoding [3]. Because of this, the decoding
algorithm is not guaranteed to find the optimal state sequenceand the training algorithm is not guaranteed to converge to a
local optimal solution. Note that this also applies to the minimum generation error (MGE) training framework [4], whichuses (10) in which V s is replaced by an identity matrix as thetraining criterion.

3. LATENT TRAJECTORY HMM
3.1. Model
While the HMM is only capable of describing piecewise stationary sequences of data vectors, the trajectory HMM is capable of describing continuously varying sequences of data
vectors, governed by discrete hidden states. This feature isnotable in that it can be used to model many kinds of timeseries data that are continuous in nature, however, it causes adifficulty as regards parameter inference. We propose introducing a conceptually similar framework based on a different
formulation, which is advantageous in that it alleviates thedifficulty related to parameter inference.
Instead of treating o as a function of c, we treat o as alatent variable that is related to c through a soft constraint o ≃W c. The relationship o ≃ W c can be expressed through theconditional distribution p(c|o). For example, we can definep(c|o) as{}1Tp(c|o) ∝ exp − (W c − o) Λ(W c − o) ,(14)2
where Λ is a constant positive definite matrix that can be set
arbitrarily. Indeed, this probability density function becomeslarger as o approaches W c. By completing the square in
the exponent of (14) with respect to c, we can write p(c|o)
explicitly asp(c|o) = N (c; mc|o , Λ−1c|o ),(15)

mc|o = Ho,

(16)

where

T
T T
Thus, p(x|s, λ) = N (x; mx , Λ−1x ) where x = [c , o ] .
−1
By using the blockwise matrix inversion formula, Λx is
given as[]
Σcc Σco−1
Σx = Λx =,(23)
Σoc ΣoowhereTΣcc = Λ−1
c|o + HU s H ,(24)

Σco = HU s ,

(25)

T

(26)
(27)

p(c|s, λ) = N (c; Hµs , Σcc ).

(28)

Σoc = U s H ,Σoo = U s .
Hence, (19) can be written as

We call this model the “latent trajectory HMM.” With thisframework, given a state sequence s and a parameter set λ, cis generated according toĉ = argmax p(c|s, λ).

(29)

c

Obviously, the solution to this is Hµs .
It is important to note that with this framework, theparameter inference problem can be dealt with using the
Expectation-Maximization (EM) algorithm by treating thejoint vector [cT , oT ]T as the complete data.
3.2. Decoding and training algorithms

H = (W T ΛW )−1 W T Λ,

(17)

Λc|o = W T ΛW .

(18)

By using this and p(o|s, λ) defined in (2), we can writep(c|s, λ) as∫p(c|s, λ) = p(c|o)p(o|s, λ)do,(19)
in a different way from (10). Geometrically, this can beviewed as a marginal distribution of the set of the projectedvalues of o onto the subspace o = W c. From (2) and (15),the joint likelihood p(c, o|s, λ) can be written asp(c, o|s, λ) = p(c|o)p(o|s, λ){([ ])T ( [ ])}1cc∝ exp −− mx Λx− mx, (20)
2
o
o
where

[
mx = Λ−1
x
[
Λx =

]

0

U −1
s µs

Λc|o
−ΛW

,

(21)

T

ŝ = argmax log p(c|s, λ)p(s)

(30)

s

{λ̂, ŝ} = argmax log p(c|s, λ)p(s).

(31)

λ,s

Since the decoding problem (30) is a subproblem of the training problem (31), here we only derive an algorithm for solving the training problem (31).
By regarding the set consisting of c and o as the complete data, this problem can be viewed as an incomplete data
problem, which can be dealt with using the ExpectationMaximization (EM) algorithm. The likelihood of s and λgiven the complete data is given by (20). By taking the conditional expectation of log p(c, o|s, λ) with respect to o givenc, s = s′ and λ = λ′ , and then adding log p(q), we obtainan auxiliary function
Q(s, λ) := Eo|c,s′,λ′ [log p(c, o|s, λ)] + log p(s).

(32)

By leaving only the terms that depend on s and λ, Q(s, λ)can be written as

]

−W Λ
.
H Λc|o H + U −1sT

As with the trajectory HMM framework, the present framework uses p(c|s, λ) for feature sequence generation, state decoding and parameter training in a consistent manner. The
problems of state decoding and parameter training can be formulated as the following optimization problems:

(22)

s,λ

Q(s, λ) = Eo|c,s′,λ′ [log p(o|s, λ)] + log p(s)

=−

(
)
1{
log |U s | + Tr U −1
s R
2
}
−1
T −1
− 2µT
s U s ō + µs U s µs + log p(s),(M-step) Update λ using (38)–(40) and findwhere′
= µ′s′ + Σ′oc Σ′−1
cc (c − Hµs′ ),(34)

R = Eo|c,s′,λ′ [ooT ]′T= Σ′oo − Σ′oc Σ′−1
cc Σco + ōō .

(35)Here, the prime mark indicates the values obtained using themodel parameters updated at the previous iteration. Since U s
is a block diagonal matrix, as given in (5), (33) can be decomposed into the sum of T individual terms:
1∑{log |U st | + Tr[U −1
st R t ]
2 t=1
T

s,λ

Q(s, λ) = −

−1
−1
T
− 2µT
st U st ōt + µst U st µst }+ log πs1 +T∑log πst−1 ,st ,

(36)

t=2

where




ō1
R1
 . 




.
ō =  .  , R = 
ōT
∗

∗
..

.



.


(37)

RT

With fixed λ, Q(s, λ) can be maximized with respect to s
by employing the Viterbi algorithm. With fixed s, Q(s, λ) is
maximized with respect to λ when∑
1[st = i]ōttµi = ∑,(38)

1[st = i]

t

∑

T
T
1[st = i](Rt − ōt µT
i − µi ōt + µi µi )t∑
πi,j =(41)s

using the Viterbi algorithm.
ō = Eo|c,s′,λ′ [o]Ui =s = argmax Q(s, λ)

(33)

t

∑

, (39)
1[st = i]3.3. Variational learning algorithm
We describe a different approach for parameter training
based on variational inference. The random variables ofinterest in our model are o, s and λ = {µ, P , π} whereµ = {µi }1≤i≤I , P = {P i := U −1i }1≤i≤I , and π ={π i,j }1≤i≤I,1≤j≤J . We denote the entire set of the above
parameters as Θ = {o, s, λ}. Our goal is to compute theposteriorp(Θ|c) =p(c, Θ)
.
p(c)

(42)

By using the conditional distributions defined in 3.1, we canwrite the joint distribution p(c, Θ) as
p(c, Θ) = p(c|o)p(o|s, λ)p(s).

(43)

To obtain the exact posterior p(Θ|c), we must compute p(c),which involves many intractable integrals. Instead of obtaining the exact posterior, we consider approximating this posterior variationally by solving an optimization problem:
argmin KL(q(Θ)∥p(Θ|c)),

t

(44)

q

1[st−1 = i, st = j]∑Note that if λ is fixed, the above algorithm reduces to a state
decoding algorithm.
It may appear that a huge amount of computation for
inverting Σ′cc is required to compute ō and R. However, this can be carried out very efficiently. First, by using the Woodbury matrix identity, Σ′−1
cc can be written as
T −1−1Λc|o −Λc|o ((HU s H ) +Λc|o ) Λc|o . Next, since Λ can
T −1
be set arbitrarily, we set Λ at U −1.
s to compute (HU s H )
T −1
T −1
Under this setting, (HU s H ) is given as W U s W .
Since both W T U −1
s W and Λc|o are sparse symmetric band
T −1matrices, (W U s W + Λc|o )−1 Λc|o can be computed
efficiently using the Cholesky decomposition.
To initialize s, one reasonable way would be to search fors = argmaxs p(o|s, λ) using the Viterbi algorithm.

,

(40)

1[st−1 = i]t∑

if t 1[st = i] ̸= 0 where i and j denote state indices and1[·] denotes an indicator function that takes the value 1 if its
argument is true and 0 otherwise.
Overall, the parameter training algorithm can be summarized as follows:(E-step) Substitute s and λ into s′ and λ′ and recompute ōand R using (34) and (35).

where KL(·∥·) denotes the Kullback-Leibler (KL) divergence
between its two arguments, i.e.,KL(q(Θ)∥p(Θ|c))∑ ∫∫q(o, s, λ)q(o, s, λ) log=
dodλ. (45)p(o,s, λ|c)s
By restricting the class of the approximate distributions tothose that factorize into independent factors:q(o, s, λ) = q(o)q(s)q(µ, P )q(π),(46)

we can use a simple coordinate ascent algorithm to find a local optimum of (44). It can be shown using the calculus ofvariations that the “optimal” distribution for each of the factors can be expressed as:q̂(X) ∝ exp EΘ\X [log p(c, Θ)],(47)

where X indicates one of the factors and EΘ\X [log p(c, Θ)]is the expectation of the joint probability of the data and latentvariables, taken over all variables except X. From (47), thevariational distributions are given in the following form:q̂(o) = N (o; m, Γ),
(48)
∏
−1
q̂(µ, P ) =N (µi ; ρi , (βi P i ) )W(P i ; B i , νi ), (49)iq̂(π i ) = Dir(π i ; αi ),(50)where the parameters are updated via the following equations


m1
 . 
−1
. 
m =
(51)
 .  ← R r,
mT


Γ1
∗


..
 ← R−1 ,
Γ =
(52)
.


∗
ΓT

∑
O
i q(s1 = i)νi B i..
 + H T Λc|o H, (53)
R =
.


∑
O
i q(sT = i)νi B i∑
i q(s1 = i)νi B i ρi..
 + H T Λc|o c,
r =
(54)
.


∑
i q(sT = i)νi B i ρi∑βi ←q(st = i),
(55)
t

1 ∑
q(st = i)mt ,
ρi ←
βi t
∑
T
B −1
←
q(st = i)(Γt + mt mTt ) − βi ρi ρi ,
i

(56)
(57)

t

νi ←βi + 3D,∑
q(st−1 = i, st = j).
αi,j ←1 +

(58)
(59)

t

W and Dir denote the Wishart distribution and the Dirichlet
distribution, respectively, defined as
W(X; V , ν) ∝ |X| 2 e− 2 Tr(V∏i −1
Dir(x; α) ∝
xα
,
i
ν−d−1

1

−1

X)

,

(60)
(61)

i

where X is a d×d symmetric matrix of random variables thatis positive definite and V is a d × d positive definite matrix.
q(st = i) and q(st−1 = i, st = j) can be computed using theforward-backward algorithm as a subroutine, as in [12].

4. EXPERIMENTS
To confirm the generalization ability of the present modeland the convergence speed of the present training algorithm, we conducted parameter training experiments using
mel-cepstrum sequences of speech as experimental data.
We chose the parameter training algorithm for the original
trajectory HMM developed by Zen et al. as the baseline
method. Zen’s algorithm uses the method of steepest ascentfor updating λ and the “delayed decision Viterbi algorithm”
for updating s. Readers are referred to [3] for the details.
Since the degrees of freedom of the trajectory HMM andthe latent trajectory HMM are exactly the same when thenumbers of hidden states are the same, the difference of the
log-likelihood scores obtained with the present and baseline
algorithms would reflect the difference in their generalization
abilities. The experimental conditions were as follows. We
used 25 speech data excerpted from the ATR speech database,from each of which we obtained the mel-cepstrum sequenceof the first 250 frames. For both the proposed model and the
trajectory HMM, the numbers of hidden states were set at 14.
Λ were fixed atΛ = diag(A, . . . , A),| {z }
T
−1
0.00010A=
00.01−1
0
0

(62)

0

0 .
0.01−1(63)

The workstation used to perform the experiments had an Intel
Core i3-2120 Processor with a 3.3GHz ×4 clock speed and a
7.7GB memory.
Fig. 2 shows the evolution of the log-likelihoods with respect to the number of iterations and computation time during
the parameter training of the proposed model and the conventional trajectory HMM. As Fig. 2 shows, the present algorithm converged faster than the conventional algorithm. Thisreveals the effectiveness of the combined use of efficient statistical inference techniques such as the EM algorithm andthe dynamic programming principle by the proposed algorithm. It is also worth noting that the converged value ofthe log-likelihood obtained with the proposed algorithm was
greater than that obtained with the conventional algorithm.
This implies the possibility that, compared with the conventional model, the proposed model has a higher generalization
ability, namely an ability to fit an arbitrary set of feature sequences, given that the degrees-of-freedom of the two models
were the same.
Fig. 3 shows an example of parameter generation using
the proposed model. After training λ using 25 speech data,a feature sequence ĉ was generated according to (29) given astate sequence s. The figure at the top shows the spectrogramof a speech sample and the figure at the bottom shows the
spectrogram constructed using ĉ obtained using the trained
λ and the state sequence s labeled from the speech sample.
As Fig. 3 shows, the proposed model was able to represent
the continuously time-varying nature of speech spectrograms
reasonably well, showing that it has a similar property to the
trajectory HMM.

800

8

600

6

200

Frequency (kHz)Log-likelihood

400

0
-200
-400
-600
-800
-1000
0

2

Proposed
Trajectory HMM
0.5

1

1.5

2

2.5

4

3

x10 4

# iterations

0
0

800

0.2

0.4

0.8

1.0

1.2

0.8

1.0

1.2

8

400
200
0

6
-200
-400
-600
Proposed
Trajectory HMM

-800
-1000
0

200

400

600

800

1000

Time (min)Fig. 2. Evolutions of the log-likelihoods with respect to the number of
iterations (top) and the computation time (bottom).
5. CONCLUSIONS
Inspired by the trajectory HMM framework proposed by Zen
et al., this paper proposed a probabilistic generative model for
describing continuously time-varying sequences of data vectors governed by discrete hidden states. The proposed modelis advantageous over the conventional trajectory HMM in thatit makes it possible to derive convergence-guaranteed and efficient algorithms for parameter training and state decoding.
Interesting future work involves incorporating the proposedmodel into the factorial HMM formulation to develop a new
method for audio source separation that takes account of the
dynamics of source spectra.
6. REFERENCES[1] T. Yoshimura, K. Tokuda, T. Masuko, T. Kobayashi, and T. Kitamura,“Simultaneous modeling of spectrum, pitch and duration in HMMbased speech synthesis,” in Proc. The 6th European Conference on
Speech Communication and Technology (EUROSPEECH 1999), 1999,
pp. 2347–2350.
[2] K. Tokuda, T. Yoshimura, T. Masuko, T. Kobayashi, and T. Kitamura,“Speech parameter generation algorithms for HMM-based speech synthesis,” in Proc. 2000 IEEE International Conference on Acoustics,
Speech and Signal Processing (ICASSP 2000), 2000, pp. 1315–1318.
[3] H. Zen, K. Tokuda, and T. Kitamura, “Reformulating the HMM as a
trajetory model by imposing explicit relationships between static anddynamic feature vector sequences,” Computer Speech and Language,
vol. 21, pp. 153–173, 2007.
[4] Y.-J. Wu and R.H. Wang, “Minimum generation error training for
HMM-based speech synthesis,” in Proc. 2006 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2006),
2006, pp. 89–92.
[5] T. Toda, A.W. Black, and K. Tokuda, “Voice conversion based on
maximum-likelihood estimation of spectral parameter trajectory,” IEEE

Frequency (kHz)Log-likelihood

0.6Time (s)

600

4

2

0
0

0.2

0.4

0.6

Time (s)Fig. 3. Example of parameter generation using the proposed model. The
spectrogram of a training sample (top) and the spectrogram constructed using
the generated parameters (bottom).

Trasanctions on Audio, Speech, and Language Processing, vol. 15, no.
8, pp. 2222–2235, 2007.
[6] T. Toda, A.W. Black, and K. Tokuda, “Statistical mapping between articulatory movements and acoustic spectrum using a Gaussian mixture
model,” Speech Communication, vol. 50, no. 3, pp. 215–227, 2007.
[7] L. Zhang and S. Renals, “Acoustic-articulatory modelling with the trajectory HMM,” IEEE Signal Processing Letters, vol. 15, pp. 245–248,
2008.
[8] M. Nakano, J. Le Roux, H. Kameoka, T. Nakamura, N. Ono, and
S. Sagayama, “Bayesian nonparametric spectrogram modeling basedon infinite factorial infinite hidden Markov model,” in Proc. 2011 IEEE
Workshop on Applications of Signal Processing to Audio and Acoustics(WASPAA2011), 2011, pp. 325–328.
[9] T. Higuchi, H. Takeda, T. Nakamura, and H. Kameoka, “A unified approach for underdetermined blind signal separation and source activity
detection by multichannel factorial hidden markov models,” in Proc.
The 15th Annual Conference of the International Speech Communication Association (Interspeech 2014), 2014, pp. 850–854.
[10] T. Higuchi and H. Kameoka, “Joint audio source separation and dereverberation based on multichannel factorial hidden markov model,” inProc. The 24th IEEE International Workshop on Machine Learning for
Signal Processing (MLSP 2014), 2014.
[11] T. Higuchi and H. Kameoka, “Unified approach for underdetermined
BSS, VAD, dereverberation and DOA estimation with multichannel
factorial HMM,” in Proc. of The 2nd IEEE Global Conference on Signal and Information Processing (GlobalSIP 2014), 2014.
[12] M.J. Beal, Variational Algorithms for Approximate Bayesian Inference, Ph.D. thesis, Gatsby Computational Neuroscience Unit, University College London, 1998.