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234 nips-2008-The Infinite Factorial Hidden Markov Model

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Author: Jurgen V. Gael, Yee W. Teh, Zoubin Ghahramani

Abstract: We introduce a new probability distribution over a potentially inﬁnite number of binary Markov chains which we call the Markov Indian buffet process. This process extends the IBP to allow temporal dependencies in the hidden variables. We use this stochastic process to build a nonparametric extension of the factorial hidden Markov model. After constructing an inference scheme which combines slice sampling and dynamic programming we demonstrate how the inﬁnite factorial hidden Markov model can be used for blind source separation. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 uk Abstract We introduce a new probability distribution over a potentially inﬁnite number of binary Markov chains which we call the Markov Indian buffet process. [sent-9, score-0.336]

2 This process extends the IBP to allow temporal dependencies in the hidden variables. [sent-10, score-0.132]

3 We use this stochastic process to build a nonparametric extension of the factorial hidden Markov model. [sent-11, score-0.421]

4 After constructing an inference scheme which combines slice sampling and dynamic programming we demonstrate how the inﬁnite factorial hidden Markov model can be used for blind source separation. [sent-12, score-0.593]

5 1 Introduction When modeling discrete time series data, the hidden Markov model [1] (HMM) is one of the most widely used and successful tools. [sent-13, score-0.166]

6 The HMM deﬁnes a probability distribution over observations y1 , y2 , · · · yT using the following generative model: it assumes there is a hidden Markov chain s1 , s2 , · · · , sT with st ∈ {1 · · · K} whose dynamics is governed by a K by K stochastic transition matrix π. [sent-14, score-0.437]

7 At each timestep t, the Markov chain generates an output yt using some likelihood model F parametrized by a state dependent parameter θst . [sent-15, score-0.306]

8 One shortcoming of the hidden Markov model is the limited representational power of the latent variables. [sent-18, score-0.239]

9 One way to look at the distribution deﬁned by the HMM is to write down the marginal distribution of yt given the previous latent state st−1 p(yt |st−1 ) = p(st |st−1 )p(yt |st ) = st πst−1 ,st F (yt ; θst ). [sent-19, score-0.374]

10 (2) st Equation (2) illustrates that the observations are generated from a dynamic mixture model. [sent-20, score-0.207]

11 The factorial hidden Markov model (FHMM), developed in [2], addresses the limited representational power of the hidden Markov model. [sent-21, score-0.505]

12 The FHMM extends the HMM by representing the hidden state ∗ http://mlg. [sent-22, score-0.164]

13 that for all our models, all latent chains start in (m) a dummy state that is in the 0 state. [sent-30, score-0.232]

14 The parallel chains can be viewed as latent features which evolve over time according to Markov dynamics. [sent-36, score-0.257]

15 Unfortunately, the dimensionality M of our factorial representation or equivalently, the number of parallel Markov chains, is a new free parameter for the FHMM which we would prefer learning from data rather than specifying it beforehand. [sent-40, score-0.238]

16 In this work, we derive the basic building block for nonparametric Bayesian factor models for time series which we call the Markov Indian Buffet Process (mIBP). [sent-44, score-0.115]

17 Using this distribution we build a nonparametric extension of the FHMM which we call the Inﬁnite Factorial Hidden Markov Model (iFHMM). [sent-45, score-0.115]

18 This construction allows us to learn a factorial representation for time series. [sent-46, score-0.251]

19 Section 4 shows results of our application of the iFHMM to a blind source separation problem. [sent-50, score-0.131]

20 In this representation rows correspond to timesteps and the columns to features or Markov chains. [sent-53, score-0.159]

21 We want the distribution over matrices to satisfy the following two properties: (1) the potential number of columns (representing latent features) should be able to be arbitrary large; (2) the rows (representing timesteps) should evolve according to a Markov process. [sent-54, score-0.246]

22 1 A ﬁnite model Let S represent a binary matrix with T rows (datapoints) and M columns (features). [sent-59, score-0.206]

23 stm represents the hidden state at time t for Markov chain m. [sent-60, score-0.299]

24 Each Markov chain evolves according to the transition matrix 1 am am W (m) = (3) 1 b m bm 2 (m) where Wij = p(st+1,m = j|stm = i). [sent-61, score-0.254]

25 We give the parameters of W (m) distributions am ∼ Beta(α/M, 1) and bm ∼ Beta(γ, δ). [sent-62, score-0.11]

26 Each chain starts with a dummy zero state s0m = 0. [sent-63, score-0.138]

27 The hidden state sequence for chain m is generated by sampling T steps from a Markov chain with transition matrix W (m) . [sent-64, score-0.37]

28 Summarizing, the generative speciﬁcation for this process is α ,1 , bm ∼ Beta(γ, δ), M s0m = 0 , stm ∼ Bernoulli(a1−st−1,m bst−1,m ). [sent-65, score-0.183]

29 m m ∀m ∈ {1, 2, · · · , M } : am ∼ Beta (4) Next, we evaluate the probability of the state matrix S with the transition matrix parameters W (m) marginalized out. [sent-66, score-0.158]

30 We introduce the following notation, let c00 , c01 , c10 , c11 be the number of 0 → m m m m 0, 0 → 1, 1 → 0 and 1 → 1 transitions respectively, in binary chain m (including the transition from the dummy state to the ﬁrst state). [sent-67, score-0.208]

31 We can then write M p(S|a, b) = 00 c01 10 c11 m m (1 − am )cm am (1 − bm )cm bm . [sent-68, score-0.22]

32 (5) m=1 We integrate out a and b with respect to the conjugate priors deﬁned in equation (4) and ﬁnd M p(S|α, γ, δ) = α α 11 10 01 00 M Γ( M + cm )Γ(cm + 1)Γ(γ + δ)Γ(δ + cm )Γ(γ + cm ) , α Γ( M + c00 + c01 + 1)Γ(γ)Γ(δ)Γ(γ + δ + c10 + c11 ) m m m m m=1 (6) where Γ(x) is the Gamma function. [sent-69, score-0.521]

33 In other words, our factorial model is exchangeable in the columns as we don’t care about the ordering of the features. [sent-74, score-0.366]

34 The left-ordered form of a binary S matrix can be deﬁned as follows: we interpret one column of length T as encoding a binary number: column m encodes the number 2T −1 s1m + 2T −2 s2m + · · · + sT m . [sent-76, score-0.178]

35 Then, we denote with Mh the number of columns in the matrix S that have the same history. [sent-78, score-0.11]

36 We say a matrix is a lofmatrix if its columns are sorted in decreasing history values. [sent-79, score-0.11]

37 p(S|α, γ, δ) (7) S∈[S] = M α α 01 00 10 11 M Γ( M + cm )Γ(cm + 1)Γ(γ + δ)Γ(δ + cm )Γ(γ + cm ) . [sent-84, score-0.48]

38 α 00 + c01 + 1)Γ(γ)Γ(δ)Γ(γ + δ + c10 + c11 ) 2T −1 Γ( M + cm m m m h=0 Mh ! [sent-85, score-0.16]

39 3 Properties of the distribution First of all, it is interesting to note from equation (9) that our model is exchangeable in the columns and Markov exchangeable2 in the rows. [sent-97, score-0.159]

40 In this stochastic process, T customers enter an Indian restaurant with an inﬁnitely long buffet of dishes organized in a line. [sent-99, score-0.297]

41 The ﬁrst customer enters the restaurant and takes a serving from each dish, starting at the left of the buffet and stopping after a Poisson(α) number of dishes as his plate becomes overburdened. [sent-100, score-0.459]

42 A waiter stands near the buffet and takes notes as to how many people have eaten which dishes. [sent-101, score-0.359]

43 The t’th customer enters the restaurant and starts at the left of the buffet. [sent-102, score-0.204]

44 At dish m, he looks at the customer in front of him to see whether he has served himself that dish. [sent-103, score-0.393]

45 The customer then serves himself dish m with probability m (c11 + δ)/(γ + δ + c10 + c11 ). [sent-105, score-0.276]

46 The customer then serves himself dish m m with probability c00 /(c00 + c01 ). [sent-107, score-0.276]

47 m m m The customer then moves on to the next dish and does exactly the same. [sent-108, score-0.276]

48 After the customer has passed all dishes people have previously served themselves from, he tries Poisson(α/t) new dishes. [sent-109, score-0.317]

49 (t) If we denote with M1 the number of new dishes tried by the t’th customer, the probability of any particular matrix being produced by this process is p([S]) = M αM+ T t=1 (t) M1 ! [sent-110, score-0.111]

50 exp{−αHT } α α 01 00 10 11 M Γ( M + cm )Γ(cm + 1)Γ(γ + δ)Γ(δ + cm )Γ(γ + cm ) . [sent-111, score-0.48]

51 4 A stick breaking representation Although the representation above is convenient for theoretical analysis, it is not very practical for inference. [sent-119, score-0.127]

52 Interestingly, we can adapt the stick breaking construction for the IBP [8] to the mIBP. [sent-120, score-0.171]

53 The ﬁrst step in the stick breaking construction is to ﬁnd the distribution of a(1) > a(2) > · · · , the order statistics of the parameters a. [sent-122, score-0.171]

54 (m−1) (m) (12) The variables bm are all independent draws from a Beta(γ, δ) distribution which is independent of M . [sent-124, score-0.11]

55 Now that we have a model with a more expressive latent structure, we want to add a likelihood model F which describes the distribution over the observations conditional on the latent structure. [sent-133, score-0.243]

56 Formally, F (yt st ) describes the probability of generating yt given the model parameters and the current latent feature state st . [sent-134, score-0.569]

57 We note that there are two important conditions which the likelihood must satisfy in order for the limit M to be valid: (1) the likelihood must be invariant to permutations of the features, (2) the likelihood cannot depend on m if stm = 0. [sent-135, score-0.225]

58 Figure 3 shows the graphical model for our construction which we call the Inﬁnite Factorial Hidden Markov Model (iFHMM). [sent-136, score-0.111]

59 As we explain in detail in section 4, we are interested in ICA to solve the blind source separation problem. [sent-141, score-0.131]

60 Assume that M signals are represented through the vectors xm ; grouping them we can represent the signals using the matrix X = [x1 x2 xM ]. [sent-142, score-0.168]

61 Next, we linearly combine the signals using a mixing matrix W to generate the observed signal Y = XW . [sent-143, score-0.181]

62 [10] used the IBP to design the Inﬁnite Independent Component Analysis (iICA) model which learns an appropriate number of signals from exchangeable data. [sent-147, score-0.155]

63 The ICA iFHMM generative model can be described as follows: we sample S mIBP and pointwise multiply (denoted by ) it with a signal matrix X. [sent-149, score-0.155]

64 One could choose many other distributions for X, but since in section 4 we will model speech data, which is known to be heavy tailed, the Laplace distribution is a convenient choice. [sent-151, score-0.135]

65 Speakers will be speaking infrequently so pointwise multiplying a heavy tailed distribution with a sparse binary matrix achieves our goal of producing a sparse heavy tailed distribution. [sent-152, score-0.312]

66 Next, we introduce a mixing matrix W which has a row for each signal in S X and a column 2 for each observed dimension in Y . [sent-153, score-0.154]

67 Finally, we combine the signal and mixing matrices as in the ﬁnite case to form the 5 2 observation matrix Y : Y = (S X)W + where is Normal(0, σY ) IID noise for each element. [sent-155, score-0.159]

68 In terms of the general iFHMM model deﬁned in the previous section, the base distribution H is a joint distribution over columns of X and rows of W . [sent-156, score-0.13]

69 The likelihood F performs the pointwise multiplication, mixes the signals and adds the noise. [sent-157, score-0.129]

70 2 Inference Inference for nonparametric models requires special treatment as the potentially unbounded dimensionality of the model makes it hard to use exact inference schemes. [sent-160, score-0.149]

71 Traditionally, in nonparametric factor models inference is done using Gibbs sampling, sometimes augmented with Metropolis Hastings steps to improve performance. [sent-161, score-0.115]

72 In the context of the inﬁnite hidden Markov model, a solution was recently proposed in [12], where a slice sampler adaptively truncates the inﬁnite dimensional model after which a dynamic programming performs exact inference. [sent-163, score-0.308]

73 Since a stick breaking construction for the iFHMM is readily available, we can use a very similar approach for the iFHMM. [sent-164, score-0.171]

74 This allows us to analytically integrate out one column of X in the dynamic program and resample it from the posterior afterwards. [sent-182, score-0.166]

75 A third algorithm runs dynamic programming on multiple chains at once. [sent-184, score-0.129]

76 6 (a) Ground Truth (b) ICA iFHMM (c) iICA (d) ICA iFHMM (e) iICA Figure 4: Blind speech separation experiment; ﬁgures represent which speaker is speaking at a certain point in time: columns are speakers, rows are white if the speaker is talking and black otherwise. [sent-190, score-0.381]

77 4 Experiments To test our model and inference algorithms, we address a blind speech separation task, also known as the cocktail party problem. [sent-192, score-0.262]

78 Given the mixed speech signals, the goal is to separate out the individual speech signals. [sent-194, score-0.128]

79 Key to our presentation is that we want to illustrate that using nonparametric methods, we can learn the number of speakers from a small amount of data. [sent-195, score-0.195]

80 Our ﬁrst experiment learns to recover the signals in a setting with more microphones then speakers, our second experiment uses less microphones then speakers. [sent-196, score-0.136]

81 The experimental setup was the following: we downloaded data from 5 speakers from the Speech Separation Challenge website3 . [sent-197, score-0.113]

82 Each mixture is a linear combination of each of the 5 speakers using Uniform(0, 1) mixing weights. [sent-201, score-0.149]

83 Visual inspection of the recovered S matrix also shows that the model discovers who is speaking at what time. [sent-221, score-0.144]

84 Although the model discovers some structure in the data, it fails to ﬁnd the right number of speakers (it ﬁnds 9) and does a poor job in discovering which speaker is active at which time. [sent-223, score-0.227]

85 We computed the average mutual information between the 5 columns of the true S matrix and the ﬁrst 5 columns of the recovered S matrices. [sent-224, score-0.214]

86 In a second experiment, we chose to perform blind speech separation using only the ﬁrst 3 microphones. [sent-230, score-0.195]

87 In this setting both methods fail to identify the number of speakers although the ICA iFHMM clearly performs better. [sent-239, score-0.113]

88 The ICA iFHMM ﬁnds one too many signal: the spurious signal is very similar to the third signal which suggests that the error is a problem of the inference algorithm and not so much of the model itself. [sent-240, score-0.145]

89 [2] introduced a factorial hidden Markov model which explicitly models dynamic latent features while in [13] a nonparametric version of the the Hidden Markov Model was presented. [sent-246, score-0.607]

90 We showed how this stochastic process can be used to build a nonparametric extension of the FHMM which we call the iFHMM. [sent-249, score-0.115]

91 Although the derivation of the mIBP with this extra parameter is straightforward, we as yet lack a stick breaking construction for this model which is crucial for our inference scheme. [sent-253, score-0.238]

92 Rabiner, “A tutorial on hidden markov models and selected applications in speech recognition,” Proceedings of the IEEE, vol. [sent-259, score-0.384]

93 Ji, “Multi-view face tracking with factorial and switching hmm,” in Proceedings of the Seventh IEEE Workshops on Application of Computer Vision, pp. [sent-270, score-0.207]

94 Duh, “Joint labeling of multiple sequences: A factorial hmm approach,” in 43rd Annual Meeting of the Association of Computational Linguistics (ACL) - Student Research Workshop, 2005. [sent-276, score-0.299]

95 Ghahramani, “Inﬁnite latent feature models and the indian buffet process,” Advances in Neural Information Processing Systems, vol. [sent-280, score-0.409]

96 Ghahramani, “Stick-breaking construction for the indian buffet process,” ou Proceedings of the International Conference on Artiﬁcial Intelligence and Statistics, vol. [sent-290, score-0.38]

97 Scott, “Bayesian methods for hidden markov models: Recursive computing in the 21st century,” Journal of the American Statistical Association, vol. [sent-304, score-0.32]

98 Ghahramani, “Beam sampling for the inﬁnite hidden markov model,” in The 25th International Conference on Machine Learning, vol. [sent-313, score-0.32]

99 Rasmussen, “The inﬁnite hidden markov model,” Advances in Neural Information Processing Systems, vol. [sent-320, score-0.32]

100 Sollich, “Bayesian nonparametric latent feature models,” Bayesian Statistics, vol. [sent-327, score-0.155]

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