Hidden Markov Model. Ying Wu. Electrical Engineering and Computer Science Northwestern University Evanston, IL 60208
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1 Hidden Markov Model Ying Wu Electrical Engineering and Computer Science Northwestern University Evanston, IL /19
2 Outline Example: Hidden Coin Tossing Hidden Markov Model Inference and Learning in HMM 2/19
3 Tossing Two Biased Coins Two biased coins (C 1 and C 2 ) Person A is selecting the coin and tossing it Person B is the observer, and is recording the sequence of H/T The selection of the coin is a secrete to B (hidden) At a certain time instance t, we have two random variables: xt indicates which coin that is chosen (C 1 or C 2 ) zt indicates the toss (H or T) The probabilities are p(x t x t 1 ) C 1 C 2 C 1 a 11 a 12 C 2 a 21 a 22 p(z t x t ) C 1 C 2 H p 1 p 2 T 1 p 1 1 p 2 p(x t x t 1 ) is a finite state machine, a discrete Markov random process. 3/19
4 Three Problems The Bayesian network...???? H T T H P1: if we have the model, and given an observation sequence, e.g., (HHTTHTTTH), determine the likelihood of this sequence P2: given the observation sequence, estimate which coin was selected at each time, i.e., infer the hidden states P3: given a number of observation sequences, estimate the parameters of the biased coins, and the parameters of the finite state machine 4/19
5 Outline Example: Hidden Coin Tossing Hidden Markov Model Inference and Learning in HMM 5/19
6 Graphical model A simple Dynamic Bayesian network... x 0 x 1 x t-1 x t Z 0 Z 1 Z t-1 Z t Hidden process x t Hidden states S = {S 1,...,S N } Observation process z t Observation symbols O = {O 1,...,O M } 6/19
7 Parameters Λ = (A,B, π) The Markov process is specified by its transition probability p(x t x t 1 ). It is a matrix A = {a ij } a ij = p(x t = S j x t 1 = S i ), 1 i, j N The observation is specified by the conditional (or emitting) prob p(z t x t ). It is a matrix B = {b j (k)} b j (k) = p(z t = O k x t = S j ), 1 j N, 1 k M p(z t x t ) S 1 S 2... S N O 1 b 1 (1) b 2 (1)... b N (1) O 2 b 1 (2) b 2 (2)... b N (2) O M b 1 (M) b 2 (M)... b N (M) The initial states (prior) π = {π i } π i = p(x 1 = S i ), 1 i N 7/19
8 A Generative Model It fully models the generation of the observation sequence S1: choose initial hidden states, and set t = 1 S2: generate observation z t = O k according to the state S i based on b i (k) S3: transit to a new state x t+1 = S j according to the dynamics S4: t t + 1, go to S2 This process generates Z t = (z 1,...,z t ) 8/19
9 Three Basic Problems P1: Likelihood: Given an observation sequence Z T = (z 1,...,z T ) and the model Λ = (A,B, π), compute the likelihood of this sequence p(z Λ) P2: Inference: Given an observation sequence Z T = (z 1,...,z T ) and the model Λ = (A,B, π), determine the hidden states X T = (x 1,...,x T ) P3: Learning: Estimate the model Λ = (A, B, π), given a set of observation sequences as training data, Z (1) T = (z(1) 1,...,z(1) T ), Z(2) T = (z(2) 1,...,z(2) T ), Z(n) T = (z(n) 1,...,z(n) T ) 9/19
10 Outline Example: Hidden Coin Tossing Hidden Markov Model Inference and Learning in HMM 10/19
11 Compute the Likelihood: the complexity The conditional likelihood T p(z T X T, Λ) = p(z t x t, Λ) t=1 = b x1 (z 1 )b x2 (z 2 )...b xt (z T ) We have p(x T Λ) = π x1 a x1 x 2 a x2 x 3...a xt 1 x t We need to sum over all possible pathes p(z T Λ) = p(z T X T, Λ)p(X T Λ) all X T = x 1,...,x T π x1 b x1 (z 1 )a x1 x 2 b x2 (z 2 )...a xt 1 x T b xt (z T ) Complexity O(2TN T ). This exhaustive method is not realistic How to make it more computationally efficient? 11/19
12 Idea: Recursion p(z 1,...,z t, x t ) = P(z }{{} 1,...,z t 1, z t, x t ) = p(z t 1, z t, x t ) α t = p(z t, x t Z t 1, x t 1 )p(z t 1, x t 1 ) x t 1 = p(z t x t ) p(x t x t 1 )p(z t 1, x t 1 ) x t 1 }{{} α t 1 p(z 1,...,z t ) = p(z 1,...,z t, x t ) = x t x t α t 12/19
13 Compute the Likelihood: Forward Algorithm Define α t (i) = p(z 1,...,z t, x t = S i Λ) = p(z t, x t = S i Λ) S1: Init: α 1 (i) = π i b i (z 1 ), 1 i N S2: Induction: [ N ] α t+1 (j) = α t (k)a kj b j (z t+1 ) k=1 S3: Termination: Complexity O(N 2 T) P(Z T Λ) = N α T (i) i=1 13/19
14 Another solution: Backward Algorithm Define Z t = (z t+1,...,z T ) Define β t (i) = p(z t+1, z t+2,...,z T x t = S i, Λ) S1: Init: β T (i) = 1, 1 i N S2: Induction: N β t (i) = a ik b k (z t+1 )β t+1 (k) S3: Termination k=1 P(Z T Λ) = N β 1 (i)π i i=1 Why does it work? Let s see the recursion p( Z t x t 1 ) = p(z t, Z t+1 x t 1 ) = p(z t, Z t+1 x t, x t 1 )p(x t x t 1 ) x t = p(z t x t )p( Z t+1 x t )p(x t x t 1 ) x t 14/19
15 To See it Clearly α t (i) and β t (i) are closely related α t (i)β t (i) = p(z 1,...,z t, x t = S i Λ)p(z t+1,...,z T x t = S i, Λ) = p(z T, x t = S i Λ) This implies (data likelihood) p(z Λ) = i α t (i)β t (i) This also implies (inference) p(x t Z, Λ) α t (i)β t (i) 15/19
16 Inferring the Hidden States: belief Define γ t (i) = p(x t = S i Z T, Λ) to be the belief of x t The MAP estimate It it clear that x t = arg max 1 i N γ t (i) = arg max p(x t = S i Z T, Λ) 1 i N γ t (i) = p(x t = S i,z T Λ) p(z T Λ) = α t (i)β t (i) N i=1 α t(i)β t (i) This is the conditional posterior of x t (i.e., the belief of x t ) It does not tell p(x 1,...,x T Z T, Λ) 16/19
17 Inferring the Hidden States: Viterbi Algorithm The MAP of the joint state is actually the optimal path Define by δ t (i) = max p(x 1,...,x t = i, z 1,...,z t Λ) x 1,...,x t 1 the best score of the single path up to time t optimal sub-path condition is satisfied [ δ t+1 (j) = max δ t (i)a ij ]b j (z t+1 ) i Construct the trellis and do dynamic programming The optimal path is obtained by backtracking 17/19
18 Learning HMM: Sufficient Statistics Define ξ t (i, j) = p(x t = S i, x t+1 = S j Z T, Λ) This is the belief of a joint pair This can be easily computed ξ t (i, j) = = p(x t = S i, x t+1 = S j,z T Λ) p(z T, Λ) α t (i)a ij b j (z t+1 )β t+1 (j) N j=1 α t(i)a ij b j (z t+1 )β t+1 (j) N i=1 It is straightforward to see the relation to γ t (i) γ t (i) = N ξ t (i, j) j=1 18/19
19 Learning: Baum-Welch Algorithm We can have an EM procedure for estimating Λ The E-step is done based on the inference process Collect the those sufficient statistics T γ t (i) expected # of transitions froms i t=1 T ξ t (i, j) expected # of transitions from S i to S j t=1 So the M-step is simply the following ˆπ i = expected freq. in S i at t = 1 = γ 1 (i) â ij = E[# of trans. from S i to S j ] T t=1 = ξ t(i, j) E[# trans. from S i ] T t=1 γ t(i) ˆb j (k) = E[# of times in S j obsrv. O k ] E[# of times in S j ] = t=i,z t=o k γ t (j) T t=1 γ t(j) 19/19
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