Hidden Markov Models
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1 Hidden Markov Models Lecture Notes Speech Communication 2, SS 2004 Erhard Rank/Franz Pernkopf Signal Processing and Speech Communication Laboratory Graz University of Technology Inffeldgasse 16c, A-8010 Graz, Austria Tel.:
2 Word Recognition Given: Word dictionary: W = {W 1,..., W L } Time-series of features from unknown word: X = {x 1,..., x N } Wanted: Most probable spoken word: W l W Target: Minimization of word error rate (WER)
3 Word Recognition/Dynamic Time Warping Alignment of observed and reference pattern
4 Word Recognition/Dynamic Time Warping Pattern matching Dynamic time warping (DTW): Dynamic programming, complexity: O(SN)
5 Word Recognition/Markov Models Word recognition by maximizing the probability of Markov model Θ l of word W l for the observed time-series of feature vectors X: l = argmax l P (Θ l X) = argmax l P (X Θ l ) P (Θ l ) P (X)
6 Markov Model/Graz Weather Today s weather Tomorrow s weather Transition probabilities: State transition diagram: Sunny Rainy 0.3 Foggy
7 Markov Model A Markov Model is specified by The set of states S = {s 1, s 2,..., s Ns }. and characterized by The prior probabilities π i = P (q 1 = s i ) Probabilities of s i being the first state of a state sequence. Collected in vector π. (The prior probabilities are often assumed equi-probable, π i = 1/N s.) The transition probabilities a ij = P (q n+1 = s j q n = s i ) probability to go from state i to state j. Collected in matrix A. The Markov model produces A state sequence Q = {q 1,..., q N }, over time 1 n N. q n S
8 Hidden Markov Model Additionally, for a Hidden Markov model we have Emission probabilities: and we get for continuous observations, e.g., x R D : b i (x) = p(x n q n = s i ) pdfs of the observation x n at time n, if the system is in state s i. Collected as a vector of functions B(x). Often parametrized, e.g, by mixtures of Gaussians. for discrete observations, x {v 1,..., v K }: b i,k = P (x n = v k q n = s i ) Probabilities for the observation of x n = v k, if the system is in state s i. Collected in matrix B. Observation sequence: X = {x 1, x 2,..., x N } HMM parameters (for fixed number of states N s ) thus are Θ = (A, B, π)
9 HMM/Graz Weather The above weather model turns into a hidden Markov model, if we can not observe the weather directly. Suppose you were locked in a room for several days, and you can only observe if a person is carrying an umbrella (v 1 = ) or not (v 2 = ). Example emission probabilities could be: Weather Probability of umbrella Sunny b 1,1 = 0.1 Rainy b 2,1 = 0.8 Foggy b 3,1 = 0.3 Since there are only two possible states for the discrete observations, the probabilities for no umbrella are b i,2 = 1 b i,
10 HMM/Discrete vs. Continuous Features Discrete features/ emission probability: HMM: Continuous features/ emission probability:
11 Markov Model/Left-to-Right Models
12 HMM/Trellis Trellis: Model description over time State 1 b 1,k a 1,1 b 1,k b 1,k b 1,k a 1,2 b 2,k b 2,k b 2,k b 2,k State 2 a 1, State 3 b 3,k b 3,k b 3,k b 3,k Sequence: x 1 x 2 x i x N n = 1 n = 2 n = i n = N time
13 HMM/Trellis example b, = 0.9 b, = 0.9 STATES a, = 0.15 b, = 0.7 a, = 0.2 Sequence: x 1 = x 2 = x 3 = n=1 n=2 n=3 time Joint likelihood for observed sequence X and state sequence (path) Q: P (X, Q Θ) = π b, a, b, a, b, = 1/
14 HMM/Parameters Parameters {π, A, B} are probabilities: positive π i 0, a i,j 0, b i,k 0 or b i (x) 0 normalization conditions N s i=1 π i = 1, N s j=1 a i,j = 1, K b i,k = 1 or k=1 X b i (x) dx =
15 Hidden Markov Models: 3 Problems The three basic problems for HMMs: Given a HMM with parameters Θ = (A, B, π), efficiently compute the production probability of an observation sequence X P (X Θ) =? (1) Given model Θ, what is the hidden state sequence Q that best explains an observation sequence X Q = argmax Q P (X, Q Θ) =? (2) How do we adjust the model parameters to maximize P (X Θ) ˆΘ = (Â, ˆB, ˆπ) =?, P (X ˆΘ) = max Θ P (X Θ) (3)
16 HMM/Prod. Probability Problem 1: Production probability Given: HMM parameters Θ Given: Observed sequence X (length N) Wanted: Probability P (X Θ), for X being produced by Θ Probability of a certain state sequence P (Q Θ) = P (q 1,..., q N Θ) = π q1 N n=2 a qn 1,q n Emission probabilities for the state sequence P (X Q, Θ) = P (x 1,..., x N q 1,..., q n, Θ) = N n=1 b qn,x n Joint probability of hidden state sequence and observation sequence N P (X, Q Θ) = P (X Q, Θ) P (Q Θ) = π q1 b q1 (x 1 ) a qn 1,q n b qn,x n n=2
17 HMM/Prod. Probability Production probability P (X Θ) = P (X, Q Θ) = Q Q N Q Q N ( π q1 b q1 (x 1 ) N n=2 a qn 1,q n b qn,x n ) Exponential complexity O(2N N N s ) use recursive algorithm (complexity linear in N N s ):
18 HMM/Prod. Probability Forward algorithm Computation of forward probabilities α n (j) = P (x 1,..., x n, q n = s j Θ) Initialization: for all j = 1... N s α 1 (j) = π i b j,x1 Recursion: for all n > 1 and all j = 1... N s ( Ns ) α n (j) = α n 1 (i) a i,j b j,xn i=1 Termination: P (X Θ) = N s j=1 α N (j)
19 HMM/Prod. Probability Backward algorithm Computation of backward probabilities β n (i) = P (x n + 1,..., x N q n = s i, Θ) Initialization: for all i = 1... N s β N (i) = 1 Recursion: for all n < N and all i = 1... N s β n (i) = N s j=1 a i,j b j,xn+1 β n+1 (j) Termination: P (X Θ) = N s π j b j,x1 β 1 (j) j=
20 HMM/Prod. Probability Forward algorithm Backward algorithm s 1 s 2 α n 1 (1) α n 1 (2) a 1,j a 2,j a i,2 a i,1 b 1,xn+1 s 1 β n+1 (1) b 2,xn+1 s 2 β n+1 (2) STATES s 3 α n 1 (3) a 3,j b j,xn s j α n (j) s i β n (i) b a 3,xn+1 i,3 s 3 β n+1 (3) a Ns,j a i,ns b Ns,x n+1 s Ns α n 1 (N s ) s Ns β n+1 (N s ) n 1 n n n + 1 time time At each time n α n (j) β n (j) = P (X, q n = s j Θ) is the joint probability of the observation sequence X and all state sequences (paths) passing through state s j at time n, and P (X Θ) = N s j=1 α n (j) β n (j)
21 HMM/State Sequence Problem 2: Hidden state sequence Given: HMM parameters Θ Given: Observed sequence X (length N) Wanted: A posteriori most probable state sequence Q Viterbi algorithm a posteriori probabilities P (Q X, Θ) = P (X, Q Θ) P (X Θ) Q is the optimal state sequence if P (X, Q Θ) = max Q Q N P (X, Q Θ) =: P (X Θ) Viterbi algorithm computes δ n (j) = max Q Q n P (x 1,..., x n, q 1,..., q n Θ) for q n = s j
22 HMM/Viterbi Algorithm Viterbi Algorithm Computation of optimal state sequence Initialization: for all j = 1... N s δ 1 (j) = π j b j,x1, ψ 1 (j) = 0 Recursion: for n > 1 and all j = 1... N s δ n (j) = max(δ n 1 a i,j ) b j,xn, i ψ n (j) = argmax(δ n 1 (i) a i,j ) i Termination: P (X Θ) = max(δ N (j)), qn = argmax(δ N (j)) j Backtracking of optimal state sequence: q n = ψ n+1 (q n+1), n = N 1, N 2,..., j
23 HMM/Viterbi Algorithm δ = δ 1 (1) δ 2 (1) δ 2 (1) δ N (1) δ 1 (2) δ 2 (2) δ 2 (2) δ N (2) δ 1 (3) δ 2 (3) δ 2 (3) δ N (3) δ 1 (4) δ 2 (4) δ 2 (4) δ N (4) ψ =???? Example: For our weather HMM Θ, find the most probable hidden weather sequence for the observation sequence X = {x 1 =, x 2 =, x 3 = } 1. Initialization (n = 1): δ 1 ( ) = π b, = 1/3 0.9 = 0.3 ψ 1 ( ) = 0 δ 1 ( ) = π b, = 1/3 0.2 = ψ 1 ( ) = 0 δ 1 ( ) = π b, = 1/3 0.7 = ψ 1 ( ) =
24 HMM/Viterbi Algorithm 2. Recursion (n = 2): We calculate the likelihood of getting to state from all possible 3 predecessor states, and choose the most likely one to go on with: δ 2 ( ) = max(δ 1 ( ) a,, δ 1 ( ) a,, δ 1 ( ) a, ) b, ψ 2 ( ) = = max( , , ) 0.1 = The likelihood is stored in δ 2, the most likely predecessor in ψ 2. The same procedure is executed with states and : δ 2 ( ) = max(δ 1 ( ) a,, δ 1 ( ) a,, δ 1 ( ) a, ) b, ψ 2 ( ) = = max( , , ) 0.8 = δ 2 ( ) = max(δ 1 ( ) a,, δ 1 ( ) a,, δ 1 ( ) a, ) b, ψ 2 ( ) = = max( , , ) 0.3 =
25 HMM/Viterbi Algorithm δ 2 ( ) = max(δ 1 ( ) a,, δ 1 ( ) a,, δ 1 ( ) a, ) b, δ 1 = 0.3 ψ 2 ( ) = STATES Sequence: x 1 = x 2 = x 3 = n = 1 n = 2 n = 3 time
26 HMM/Viterbi Algorithm Recursion (n = 3): δ 3 ( ) = max(δ 2 ( ) a,, δ 2 ( ) a,, δ 2 ( ) a, ) b, = max( , , ) 0.1 = ψ 3 ( ) = δ 3 ( ) = max(δ 2 ( ) a,, δ 2 ( ) a,, δ 2 ( ) a, ) b, = max( , , ) 0.8 = ψ 3 ( ) = δ 3 ( ) = max(δ 2 ( ) a,, δ 2 ( ) a,, δ 2 ( ) a, ) b, = max( , , ) 0.3 = ψ 3 ( ) =
27 HMM/Viterbi Algorithm δ 3 ( ) = STATES δ 3 ( ) = δ 3 ( ) = Sequence: x 1 = x 2 = x 3 = n = 1 n = 2 n = 3 time
28 HMM/Viterbi Algorithm 3. Termination The globally most likely path is determined, starting by looking for the last state of the most likely sequence. P (X Θ) = max(δ 3 (i)) = δ 3 ( ) = q 3 = argmax(δ 3 (i)) = 4. Backtracking The best sequence of states can be read from the ψ vectors. n = N 1 = 2: q 2 = ψ 3 (q 3) = ψ 3 ( ) = n = N 1 = 1: q 1 = ψ 2 (q 2) = ψ 2 ( ) = The most likely weather sequence is: Q = {q 1, q 2, q 3} = {,, }
29 HMM/Viterbi Algorithm Backtracking: δ 3 ( ) = STATES δ 3 ( ) = δ 3 ( ) = Sequence: x 1 = x 2 = x 3 = n = 1 n = 2 n = 3 time
30 HMM/Parameter Estimation Problem 3: Parameter estimation for HMMs Given: HMM structure (N s states, K observation symbols) Given: Training sequence X = {x 1,..., x N } Wanted: optimal parameter values ˆΘ = {ˆπ, Â, ˆB} P (X ˆΘ) = max Θ P (X Θ) = max Θ Q Q N P (X, Q ˆΘ) Baum-Welch Algorithm or EM (Expectation-Maximization) Algorithm Iterative optimization of parameters Θ ˆΘ In the terminology of the EM algorithm we have observable variables: observation sequence X hyper-parameters: state sequence Q
31 HMM/Parameter Estimation Transition probabilities for s i s j at time n (for given Θ): ξ n (i, j) := P (q n = s i, q n+1 = s j X, Θ) = α n(i) a i,j b j,xn+1 β n+1 (j) P (X Θ) b j,xn+1 STATES s i α n (i) a i,j s j β n+1 (j) n 1 n n + 1 n + 2 State probability for s i at time n (for given Θ): γ n (i) := P (q n = s i X, Θ) = α n(i) β n (i) P (X Θ) P (X Θ) = N s α n (i) β n (i) = time N s j=1 ξ n (i, j) (cf. forward/backward algorithm) 2004 i=1 31
32 HMM/Parameter Estimation Summing over time n gives expected numbers # (frequencies) for N γ n (i)... # of transitions from state s i n=1 N ξ n (i, j)... # of transitions from state s i to state s j n=1 Baum-Welch update of HMM parameters: π i = γ 1 (i)...# of state s i at time n = 1 ā i,j = bj,k = N 1 n=1 ξ n(i, j) N 1 n=1 γ n(i, j) N n=1 γ n(i, j) [x n = v k ] N n=1 γ n(i, j)... # of transitions from state s i to state s j # of transitions from state s i... # of state s i with v k emitted # of state s i
33 HMM/Parameter Estimation Gaussian (normal distributed) emission probabilities: b j (x) = N (x µ j, Σ j ) Mixtures of Gaussians b j (x) = K k=1 c jk N (x µ jk, Σ jk ), Semi-continuous emission probabilities: b j (x) = K k=1 c jk N (x µ k, Σ k ), K k=1 c jk = 1 K k=1 c jk =
34 HMM/Parameter Estimation Problems encountered for HMM parameter estimation many word models/hmm states/parameters... always too less training data! Consequences: large variance of estimated parameters large variance in objective function P (X Θ) vanishing statistics zero valued parameters â i,j, ˆb j,k, ˆΣ k, ˆΣ jk,... Possible remedies (besides using more training data): fix some parameter values tying parameter values for similar models interpolation of sensible parameters by robust parameters smoothing of probability density functions defining limits for sensible density parameters
35 HMM/Parameter Estimation Parameter tying simultaneous identification of parameters for similar models forces identical parameter values reduces parameter space dimension Example (state tying): Automatic determination of states that can be tied, e.g., by mutual information
36 HMM/Parameter Estimation Parameter interpolation instead of fixed tying of states: interpolate parameters of similar models P (X Θ R, Θ S, r R, r S ) = r R P (X Θ R ) + r S P (X Θ S ), r R + r S = 1 especially suited for semi-continuous emission pdfs weights r R, r S can be chosen heuristically or included in the Baum-Welch algorithm
37 References R.O. Duda and P.E. Hart, Pattern Classification and Scene Analysis. Wiley&Sons, Inc., S. Bengio, An Introduction to Statistical Machine Learning EM for GMMs, Dalle Molle Institute for Perceptual Artificial Intelligence. E.G. Schukat-Talamazzini, Automatische Spracherkennung, Vieweg-Verlag, L.R. Rabiner, A tutorial on hidden Markov models and selected applications in speech recognition, Proceedings of the IEEE, Vol. 77, No. 2, pp ,
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