PRML 13.2.2: The Forward-Backward Algorithm

1. PRML 13.2.2 The forward-backward algorithm August 4, 2014 by Shinichi TAMURA

2. The forward-backward algorithm Today's topics The forward-backward algorithm 1.  E-step of EM algorithm for HMM 2.  Multiple shorter sequences 3.  Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA

5. The forward-backward algorithm E-step of EM algorithm for HMM Quick review HMM is trained by EM algorithm E-step: Evaluate γ(zn) and ξ(zn-1, zn) M-step: Update parameters Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA πk = γ(z1k) K j=1 γ(z1j) , Ajk = N n=2 ξ(zn−1, j, znk) K l=1 N n=2 ξ(zn−1, j, znl) .

6. The forward-backward algorithm E-step of EM algorithm for HMM Quick review HMM is trained by EM algorithm E-step: Evaluate γ(zn) and ξ(zn-1, zn) M-step: Update parameters Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA πk = γ(z1k) K j=1 γ(z1j) , Ajk = N n=2 ξ(zn−1, j, znk) K l=1 N n=2 ξ(zn−1, j, znl) .

7. The forward-backward algorithm E-step of EM algorithm for HMM Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn-1, zn) γ(zn)

8. The forward-backward algorithm E-step of EM algorithm for HMM Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn-1, zn) γ(zn) ? ?

9. The forward-backward algorithm E-step of EM algorithm for HMM How to calculate γ(zn), ξ(zn-1, zn) ? Basic idea is message passing on tree (It will be studied later, and we shall do conventional style now) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA

10. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of γ(zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA γ(zn) = p(zn|x1:N) = p(x1:N|zn)p(zn) p(x1:N) = p(x1:n|zn)p(zn)p(xn+1:N|zn) p(x1:N)

13. The forward-backward algorithm E-step of EM algorithm for HMM Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA zn+1 zNzn-1 znz1 xn+1 xNxn-1 xnx1 Head Tail Tail

16. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of γ(zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA γ(zn) = p(zn|x1:N) = p(x1:N|zn)p(zn) p(x1:N) = p(x1:n|zn)p(zn)p(xn+1:N|zn) p(x1:N) β(zn) α(zn)

17. The forward-backward algorithm E-step of EM algorithm for HMM Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn-1, zn) γ(zn) ? ?

18. The forward-backward algorithm E-step of EM algorithm for HMM Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn-1, zn) γ(zn) β(zn) α(zn) ? ? ? p(x1:N) ?

19. The forward-backward algorithm E-step of EM algorithm for HMM Now problem is changed into How to calculate α(zn), β(zn) ? Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA

22. The forward-backward algorithm E-step of EM algorithm for HMM Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA zn-1 znz1 xn-1 xnx1 Head Tail

28. The forward-backward algorithm E-step of EM algorithm for HMM Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA zn-1 znz1 xn-1 xnx1 Head Tail Tail

33. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of α(zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA α(zn) = p(xn|zn) zn−1 α(zn−1)A. k = 1 k = 2 k = K ... n = 1 n = 2 n = N... O(K)

34. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of α(zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA α(zn) = p(xn|zn) zn−1 α(zn−1)A. k = 1 k = 2 k = K ... n = 1 n = 2 n = N... O(K2 )

35. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of α(zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA α(zn) = p(xn|zn) zn−1 α(zn−1)A. k = 1 k = 2 k = K ... n = 1 n = 2 n = N... O(K2 N)

36. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of α(zn) Start of recursion? Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA α(z1) = p(x1|z1)p(z1) =   π1 p(x1|φ1) ... πN p(x1|φN)   .

37. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of α(zn) Start of recursion? Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA α(z1) = p(x1|z1)p(z1) =   π1 p(x1|φ1) ... πN p(x1|φN)   .

38. The forward-backward algorithm E-step of EM algorithm for HMM Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn-1, zn) γ(zn) β(zn) α(zn) ? ? ? p(x1:N) ?

39. The forward-backward algorithm E-step of EM algorithm for HMM Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn-1, zn) γ(zn) β(zn) α(zn) ? ? p(x1:N) ?

43. The forward-backward algorithm E-step of EM algorithm for HMM Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA zn+2 zNzn zn+1 xn+2 xNxn xn+1 Head Tail

47. The forward-backward algorithm E-step of EM algorithm for HMM Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA zn+2 zNzn zn+1 xn+2 xNxn xn+1 Tail Tail

52. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of β(zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA β(zn) = zn+1 β(zn+1)p(xn+1|zn+1)AT . k = 1 k = 2 k = K ... n = 1 n = 2 n = N... O(K2 N)

53. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of β(zn) Start of recursion? Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA p(zN|x1:N) = p(x1:N|zN)p(zN)β(zN) p(x1:N) ⇔ β(zN) = 1 · · · 1 T .

54. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of β(zn) Start of recursion? Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA p(zN|x1:N) = p(x1:N|zN)p(zN)β(zN) p(x1:N) ⇔ β(zN) = 1 · · · 1 T .

55. The forward-backward algorithm E-step of EM algorithm for HMM Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn-1, zn) γ(zn) β(zn) α(zn) ? ? p(x1:N) ?

56. The forward-backward algorithm E-step of EM algorithm for HMM Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn-1, zn) γ(zn) β(zn) α(zn) ? p(x1:N) ?

57. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of p(x1:N) It appears on the denominator of γ(zn). Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA γ(zn) = α(zn)β(zn) p(x1:N)

58. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of p(x1:N) It appears on the denominator of γ(zn). In fact, we don't need it for update because it cancel out. Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA πk = γ(z1k) K j=1 γ(z1j) , Ajk = N n=2 ξ(zn−1, j, znk) K l=1 N n=2 ξ(zn−1, j, znl) .

59. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of p(x1:N) It appears on the denominator of γ(zn). In fact, we don't need it for update because it cancel out. However, we need to evaluate it, because it's LIKELIHOOD, which is monitored. Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA

60. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of p(x1:N) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA γ(zn) = α(zn)β(zn) p(X) . ⇔ zn γ(zn) = zn α(zn)β(zn) p(X) . ⇔ p(X) = zn α(zn)β(zn) ( γ = 1). ∴ p(X) = zN α(zN) = k αk(zN).

62. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of p(x1:N) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA γ(zn) = α(zn)β(zn) p(X) . ⇔ zn γ(zn) = zn α(zn)β(zn) p(X) . ⇔ p(X) = zn α(zn)β(zn) ( γ = 1). ∴ p(X) = zN α(zN) = k αk(zN).Any n will do

64. The forward-backward algorithm E-step of EM algorithm for HMM Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn-1, zn) γ(zn) β(zn) α(zn) ? p(x1:N) ?

65. The forward-backward algorithm E-step of EM algorithm for HMM Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn-1, zn) γ(zn) β(zn) α(zn) ? p(x1:N) Nth

69. The forward-backward algorithm E-step of EM algorithm for HMM Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA zn+1 zNzn-1 znz1 xn+1 xNxn-1 xnx1 Head Tail TailTail

73. The forward-backward algorithm E-step of EM algorithm for HMM Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA zn+1 zNzn-1 znz1 xn+1 xNxn-1 xnx1 Head Tail Tail

77. The forward-backward algorithm E-step of EM algorithm for HMM Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA zn+1 zNzn-1 znz1 xn+1 xNxn-1 xnx1 Head Tail

83. The forward-backward algorithm E-step of EM algorithm for HMM Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA zn+1 zNzn-1 znz1 xn+1 xNxn-1 xnx1 Head Tail

88. The forward-backward algorithm E-step of EM algorithm for HMM Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn-1, zn) γ(zn) β(zn) α(zn) ? p(x1:N) Nth

89. The forward-backward algorithm E-step of EM algorithm for HMM Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn-1, zn) γ(zn) β(zn) α(zn) p(x1:N) Nth

90. The forward-backward algorithm E-step of EM algorithm for HMM Summary HMM is trained by EM algorithm E-step: Evaluate α(zn) and β(zn) M-step: Update parameters Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA α(zn) = p(xn|zn) zn−1 α(zn−1)p(zn|zn−1), β(zn) = zn+1 β(zn+1)p(xn+1|zn+1)p(zn+1|zn).

93. The forward-backward algorithm Multiple shorter sequences In some practical situation, we can't get a long sequence. Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA

94. The forward-backward algorithm Multiple shorter sequences In some practical situation, we can't get a long sequence. Instead, we get multiple shorter sequences. Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA

95. The forward-backward algorithm Multiple shorter sequences In some practical situation, we can't get a long sequence. Instead, we get multiple shorter sequences. In this situation, we are still able to use forward-backward algorithm (w/ bit modiﬁcation). Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA

96. The forward-backward algorithm Multiple shorter sequences E-step Just evaluate independently Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA γ(z(r) n ) = p(z(r) n |X(r) , θold ), ξ(z(r) n−1, z(r) n ) = p(z(r) n−1, z(r) n |X(r) , θold ).

97. The forward-backward algorithm Multiple shorter sequences E-step Just evaluate independently Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA γ(z(r) n ) = p(z(r) n |X(r) , θold ), ξ(z(r) n−1, z(r) n ) = p(z(r) n−1, z(r) n |X(r) , θold ).

98. The forward-backward algorithm Multiple shorter sequences M-step Just add them all etc. Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA πk = R r=1 γ(z(r) 1k ) R r=1 K j=1 γ(z(r) i j ) , Ajk = R r=1 Nr n=2 ξ(z(r) n−1, j, z(r) nk ) R r=1 K l=1 Nr n=2 ξ(z(r) n−1, j, z(r) nl )

99. The forward-backward algorithm Multiple shorter sequences M-step Just add them all etc. Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA πk = R r=1 γ(z(r) 1k ) R r=1 K j=1 γ(z(r) i j ) , Ajk = R r=1 Nr n=2 ξ(z(r) n−1, j, z(r) nk ) R r=1 K l=1 Nr n=2 ξ(z(r) n−1, j, z(r) nl )

100. The forward-backward algorithm Multiple shorter sequences M-step Just add them all etc. Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA πk = R r=1 γ(z(r) 1k ) R r=1 K j=1 γ(z(r) i j ) , Ajk = R r=1 Nr n=2 ξ(z(r) n−1, j, z(r) nk ) R r=1 K l=1 Nr n=2 ξ(z(r) n−1, j, z(r) nl ) Note: p(X) no longer cancel out

103. The forward-backward algorithm Predictive distribution Once you trained the HMM, next thing you want to do may be PREDICTION. I.e., evaluation of p(xN+1|X) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA

107. The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA zN zN+1z1 xN xN+1x1 Head Tail

114. The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA zN zN+1z1 xN xN+1x1 Head Tail Tail

120. The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn-1, zn) γ(zn) β(zn) α(zn) p(x1:N) Nth

121. The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn-1, zn) γ(zn) p(x1:N) β(zn) α(zn) p(xN+1|X) Nth Nth

PRML 13.2.2: The Forward-Backward Algorithm

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