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PRML 13.2.2: The Forward-Backward Algorithm

Shinichi Tamura, profile picture
Shinichi Tamura

The document discusses the forward-backward algorithm, emphasizing its role as the E-step of the EM algorithm for Hidden Markov Models (HMM). It covers the evaluation of the parameters γ(zn) and ξ(zn-1, zn) and provides equations for calculating these parameters. The algorithm utilizes message passing concepts and is further illustrated through detailed equations and evaluations for α(zn) and β(zn).

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PRML 13.2.2 The forward-backward algorithm August 4, 2014 by Shinichi TAMURA
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
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
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
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) .
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) .
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)
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) ? ?
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
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)
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)
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)
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
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)
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)
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)
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) ? ?
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) ?
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
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(x1:n|zn)p(zn) = p(xn|zn)p(x1:n−1|zn)p(zn) = p(xn|zn)p(x1:n−1, zn) = p(xn|zn) zn−1 p(x1:n−1, zn|zn−1)p(zn−1) = p(xn|zn) zn−1 p(x1:n−1|zn−1)p(zn|zn−1)p(zn−1) = p(xn|zn) zn−1 α(zn−1)A.
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(x1:n|zn)p(zn) = p(xn|zn)p(x1:n−1|zn)p(zn) = p(xn|zn)p(x1:n−1, zn) = p(xn|zn) zn−1 p(x1:n−1, zn|zn−1)p(zn−1) = p(xn|zn) zn−1 p(x1:n−1|zn−1)p(zn|zn−1)p(zn−1) = p(xn|zn) zn−1 α(zn−1)A.
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
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(x1:n|zn)p(zn) = p(xn|zn)p(x1:n−1|zn)p(zn) = p(xn|zn)p(x1:n−1, zn) = p(xn|zn) zn−1 p(x1:n−1, zn|zn−1)p(zn−1) = p(xn|zn) zn−1 p(x1:n−1|zn−1)p(zn|zn−1)p(zn−1) = p(xn|zn) zn−1 α(zn−1)A.
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(x1:n|zn)p(zn) = p(xn|zn)p(x1:n−1|zn)p(zn) = p(xn|zn)p(x1:n−1, zn) = p(xn|zn) zn−1 p(x1:n−1, zn|zn−1)p(zn−1) = p(xn|zn) zn−1 p(x1:n−1|zn−1)p(zn|zn−1)p(zn−1) = p(xn|zn) zn−1 α(zn−1)A.
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(x1:n|zn)p(zn) = p(xn|zn)p(x1:n−1|zn)p(zn) = p(xn|zn)p(x1:n−1, zn) = p(xn|zn) zn−1 p(x1:n−1, zn|zn−1)p(zn−1) = p(xn|zn) zn−1 p(x1:n−1|zn−1)p(zn|zn−1)p(zn−1) = p(xn|zn) zn−1 α(zn−1)A.
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(x1:n|zn)p(zn) = p(xn|zn)p(x1:n−1|zn)p(zn) = p(xn|zn)p(x1:n−1, zn) = p(xn|zn) zn−1 p(x1:n−1, zn|zn−1)p(zn−1) = p(xn|zn) zn−1 p(x1:n−1|zn−1)p(zn|zn−1)p(zn−1) = p(xn|zn) zn−1 α(zn−1)A.
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(x1:n|zn)p(zn) = p(xn|zn)p(x1:n−1|zn)p(zn) = p(xn|zn)p(x1:n−1, zn) = p(xn|zn) zn−1 p(x1:n−1, zn|zn−1)p(zn−1) = p(xn|zn) zn−1 p(x1:n−1|zn−1)p(zn|zn−1)p(zn−1) = p(xn|zn) zn−1 α(zn−1)A.
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
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(x1:n|zn)p(zn) = p(xn|zn)p(x1:n−1|zn)p(zn) = p(xn|zn)p(x1:n−1, zn) = p(xn|zn) zn−1 p(x1:n−1, zn|zn−1)p(zn−1) = p(xn|zn) zn−1 p(x1:n−1|zn−1)p(zn|zn−1)p(zn−1) = p(xn|zn) zn−1 α(zn−1)A.
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(x1:n|zn)p(zn) = p(xn|zn)p(x1:n−1|zn)p(zn) = p(xn|zn)p(x1:n−1, zn) = p(xn|zn) zn−1 p(x1:n−1, zn|zn−1)p(zn−1) = p(xn|zn) zn−1 p(x1:n−1|zn−1)p(zn|zn−1)p(zn−1) = p(xn|zn) zn−1 α(zn−1)A.
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(x1:n|zn)p(zn) = p(xn|zn)p(x1:n−1|zn)p(zn) = p(xn|zn)p(x1:n−1, zn) = p(xn|zn) zn−1 p(x1:n−1, zn|zn−1)p(zn−1) = p(xn|zn) zn−1 p(x1:n−1|zn−1)p(zn|zn−1)p(zn−1) = p(xn|zn) zn−1 α(zn−1)A.
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(x1:n|zn)p(zn) = p(xn|zn)p(x1:n−1|zn)p(zn) = p(xn|zn)p(x1:n−1, zn) = p(xn|zn) zn−1 p(x1:n−1, zn|zn−1)p(zn−1) = p(xn|zn) zn−1 p(x1:n−1|zn−1)p(zn|zn−1)p(zn−1) = p(xn|zn) zn−1 α(zn−1)A.
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)
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 )
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)
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)   .
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)   .
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) ?
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) ?
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) = p(xn+1:N|zn) = zn+1 p(xn+1:N|zn, zn+1)p(zn+1|zn) = zn+1 p(xn+1:N|zn+1)p(zn+1|zn) = zn+1 p(xn+2:N|zn+1)p(xn+1|zn+1)p(zn+1|zn) = zn+1 β(zn+1)p(xn+1|zn+1)AT .
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) = p(xn+1:N|zn) = zn+1 p(xn+1:N|zn, zn+1)p(zn+1|zn) = zn+1 p(xn+1:N|zn+1)p(zn+1|zn) = zn+1 p(xn+2:N|zn+1)p(xn+1|zn+1)p(zn+1|zn) = zn+1 β(zn+1)p(xn+1|zn+1)AT .
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) = p(xn+1:N|zn) = zn+1 p(xn+1:N|zn, zn+1)p(zn+1|zn) = zn+1 p(xn+1:N|zn+1)p(zn+1|zn) = zn+1 p(xn+2:N|zn+1)p(xn+1|zn+1)p(zn+1|zn) = zn+1 β(zn+1)p(xn+1|zn+1)AT .
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
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) = p(xn+1:N|zn) = zn+1 p(xn+1:N|zn, zn+1)p(zn+1|zn) = zn+1 p(xn+1:N|zn+1)p(zn+1|zn) = zn+1 p(xn+2:N|zn+1)p(xn+1|zn+1)p(zn+1|zn) = zn+1 β(zn+1)p(xn+1|zn+1)AT .
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) = p(xn+1:N|zn) = zn+1 p(xn+1:N|zn, zn+1)p(zn+1|zn) = zn+1 p(xn+1:N|zn+1)p(zn+1|zn) = zn+1 p(xn+2:N|zn+1)p(xn+1|zn+1)p(zn+1|zn) = zn+1 β(zn+1)p(xn+1|zn+1)AT .
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) = p(xn+1:N|zn) = zn+1 p(xn+1:N|zn, zn+1)p(zn+1|zn) = zn+1 p(xn+1:N|zn+1)p(zn+1|zn) = zn+1 p(xn+2:N|zn+1)p(xn+1|zn+1)p(zn+1|zn) = zn+1 β(zn+1)p(xn+1|zn+1)AT .
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
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) = p(xn+1:N|zn) = zn+1 p(xn+1:N|zn, zn+1)p(zn+1|zn) = zn+1 p(xn+1:N|zn+1)p(zn+1|zn) = zn+1 p(xn+2:N|zn+1)p(xn+1|zn+1)p(zn+1|zn) = zn+1 β(zn+1)p(xn+1|zn+1)AT .
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) = p(xn+1:N|zn) = zn+1 p(xn+1:N|zn, zn+1)p(zn+1|zn) = zn+1 p(xn+1:N|zn+1)p(zn+1|zn) = zn+1 p(xn+2:N|zn+1)p(xn+1|zn+1)p(zn+1|zn) = zn+1 β(zn+1)p(xn+1|zn+1)AT .
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) = p(xn+1:N|zn) = zn+1 p(xn+1:N|zn, zn+1)p(zn+1|zn) = zn+1 p(xn+1:N|zn+1)p(zn+1|zn) = zn+1 p(xn+2:N|zn+1)p(xn+1|zn+1)p(zn+1|zn) = zn+1 β(zn+1)p(xn+1|zn+1)AT .
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) = p(xn+1:N|zn) = zn+1 p(xn+1:N|zn, zn+1)p(zn+1|zn) = zn+1 p(xn+1:N|zn+1)p(zn+1|zn) = zn+1 p(xn+2:N|zn+1)p(xn+1|zn+1)p(zn+1|zn) = zn+1 β(zn+1)p(xn+1|zn+1)AT .
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)
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 .
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 .
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) ?
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) ?
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)
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) .
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
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).
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).
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
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).
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) ?
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
The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
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
The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
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
The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
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
The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn|zn−1)p(zn−1) p(x1:N) = α(zn−1)p(xn|zn)p(zn|zn−1)β(zn) p(x1:N) .
The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn|zn−1)p(zn−1) p(x1:N) = α(zn−1)p(xn|zn)p(zn|zn−1)β(zn) p(x1:N) .
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
The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn|zn−1)p(zn−1) p(x1:N) = α(zn−1)p(xn|zn)p(zn|zn−1)β(zn) p(x1:N) .
The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn|zn−1)p(zn−1) p(x1:N) = α(zn−1)p(xn|zn)p(zn|zn−1)β(zn) p(x1:N) .
The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn|zn−1)p(zn−1) p(x1:N) = α(zn−1)p(xn|zn)p(zn|zn−1)β(zn) p(x1:N) . β(zn) α(zn)
The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn|zn−1)p(zn−1) p(x1:N) = α(zn−1)p(xn|zn)p(zn|zn−1)β(zn) p(x1:N) .
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
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
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).
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
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
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
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
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
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 ).
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 ).
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 )
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 )
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
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
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
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
The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA p(xN+1|X) = zN+1 p(xN+1, zN+1|X) = zN+1 p(xN+1|zN+1, X)p(zN+1|X) = zN+1 p(xN+1|zN+1)p(zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN, zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) ...
The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA p(xN+1|X) = zN+1 p(xN+1, zN+1|X) = zN+1 p(xN+1|zN+1, X)p(zN+1|X) = zN+1 p(xN+1|zN+1)p(zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN, zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) ...
The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA p(xN+1|X) = zN+1 p(xN+1, zN+1|X) = zN+1 p(xN+1|zN+1, X)p(zN+1|X) = zN+1 p(xN+1|zN+1)p(zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN, zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) ...
The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA zN zN+1z1 xN xN+1x1 Head Tail
The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA p(xN+1|X) = zN+1 p(xN+1, zN+1|X) = zN+1 p(xN+1|zN+1, X)p(zN+1|X) = zN+1 p(xN+1|zN+1)p(zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN, zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) ...
The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA p(xN+1|X) = zN+1 p(xN+1, zN+1|X) = zN+1 p(xN+1|zN+1, X)p(zN+1|X) = zN+1 p(xN+1|zN+1)p(zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN, zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) ...
The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA p(xN+1|X) = zN+1 p(xN+1, zN+1|X) = zN+1 p(xN+1|zN+1, X)p(zN+1|X) = zN+1 p(xN+1|zN+1)p(zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN, zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) ...
The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA p(xN+1|X) = zN+1 p(xN+1, zN+1|X) = zN+1 p(xN+1|zN+1, X)p(zN+1|X) = zN+1 p(xN+1|zN+1)p(zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN, zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) ...
The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN) p(zN, X) p(X) = 1 p(X) zN+1 p(xN+1|zN+1) zN p(zN+1|zN)α(zN).
The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN) p(zN, X) p(X) = 1 p(X) zN+1 p(xN+1|zN+1) zN p(zN+1|zN)α(zN).
The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA zN zN+1z1 xN xN+1x1 Head Tail Tail
The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN) p(zN, X) p(X) = 1 p(X) zN+1 p(xN+1|zN+1) zN p(zN+1|zN)α(zN).
The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN) p(zN, X) p(X) = 1 p(X) zN+1 p(xN+1|zN+1) zN p(zN+1|zN)α(zN).
The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN) p(zN, X) p(X) = 1 p(X) zN+1 p(xN+1|zN+1) zN p(zN+1|zN)α(zN).
The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN) p(zN, X) p(X) = 1 p(X) zN+1 p(xN+1|zN+1) zN p(zN+1|zN)α(zN).
The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN) p(zN, X) p(X) = 1 p(X) zN+1 p(xN+1|zN+1) zN p(zN+1|zN)α(zN). Only depend on α(zN) i.e., whole X don't need
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
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
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
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
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

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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
  • 3. 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
  • 4. 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)
  • 11. 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)
  • 12. 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
  • 14. 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)
  • 15. 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)
  • 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
  • 20. 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(x1:n|zn)p(zn) = p(xn|zn)p(x1:n−1|zn)p(zn) = p(xn|zn)p(x1:n−1, zn) = p(xn|zn) zn−1 p(x1:n−1, zn|zn−1)p(zn−1) = p(xn|zn) zn−1 p(x1:n−1|zn−1)p(zn|zn−1)p(zn−1) = p(xn|zn) zn−1 α(zn−1)A.
  • 21. 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(x1:n|zn)p(zn) = p(xn|zn)p(x1:n−1|zn)p(zn) = p(xn|zn)p(x1:n−1, zn) = p(xn|zn) zn−1 p(x1:n−1, zn|zn−1)p(zn−1) = p(xn|zn) zn−1 p(x1:n−1|zn−1)p(zn|zn−1)p(zn−1) = p(xn|zn) zn−1 α(zn−1)A.
  • 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
  • 23. 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(x1:n|zn)p(zn) = p(xn|zn)p(x1:n−1|zn)p(zn) = p(xn|zn)p(x1:n−1, zn) = p(xn|zn) zn−1 p(x1:n−1, zn|zn−1)p(zn−1) = p(xn|zn) zn−1 p(x1:n−1|zn−1)p(zn|zn−1)p(zn−1) = p(xn|zn) zn−1 α(zn−1)A.
  • 24. 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(x1:n|zn)p(zn) = p(xn|zn)p(x1:n−1|zn)p(zn) = p(xn|zn)p(x1:n−1, zn) = p(xn|zn) zn−1 p(x1:n−1, zn|zn−1)p(zn−1) = p(xn|zn) zn−1 p(x1:n−1|zn−1)p(zn|zn−1)p(zn−1) = p(xn|zn) zn−1 α(zn−1)A.
  • 25. 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(x1:n|zn)p(zn) = p(xn|zn)p(x1:n−1|zn)p(zn) = p(xn|zn)p(x1:n−1, zn) = p(xn|zn) zn−1 p(x1:n−1, zn|zn−1)p(zn−1) = p(xn|zn) zn−1 p(x1:n−1|zn−1)p(zn|zn−1)p(zn−1) = p(xn|zn) zn−1 α(zn−1)A.
  • 26. 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(x1:n|zn)p(zn) = p(xn|zn)p(x1:n−1|zn)p(zn) = p(xn|zn)p(x1:n−1, zn) = p(xn|zn) zn−1 p(x1:n−1, zn|zn−1)p(zn−1) = p(xn|zn) zn−1 p(x1:n−1|zn−1)p(zn|zn−1)p(zn−1) = p(xn|zn) zn−1 α(zn−1)A.
  • 27. 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(x1:n|zn)p(zn) = p(xn|zn)p(x1:n−1|zn)p(zn) = p(xn|zn)p(x1:n−1, zn) = p(xn|zn) zn−1 p(x1:n−1, zn|zn−1)p(zn−1) = p(xn|zn) zn−1 p(x1:n−1|zn−1)p(zn|zn−1)p(zn−1) = p(xn|zn) zn−1 α(zn−1)A.
  • 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
  • 29. 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(x1:n|zn)p(zn) = p(xn|zn)p(x1:n−1|zn)p(zn) = p(xn|zn)p(x1:n−1, zn) = p(xn|zn) zn−1 p(x1:n−1, zn|zn−1)p(zn−1) = p(xn|zn) zn−1 p(x1:n−1|zn−1)p(zn|zn−1)p(zn−1) = p(xn|zn) zn−1 α(zn−1)A.
  • 30. 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(x1:n|zn)p(zn) = p(xn|zn)p(x1:n−1|zn)p(zn) = p(xn|zn)p(x1:n−1, zn) = p(xn|zn) zn−1 p(x1:n−1, zn|zn−1)p(zn−1) = p(xn|zn) zn−1 p(x1:n−1|zn−1)p(zn|zn−1)p(zn−1) = p(xn|zn) zn−1 α(zn−1)A.
  • 31. 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(x1:n|zn)p(zn) = p(xn|zn)p(x1:n−1|zn)p(zn) = p(xn|zn)p(x1:n−1, zn) = p(xn|zn) zn−1 p(x1:n−1, zn|zn−1)p(zn−1) = p(xn|zn) zn−1 p(x1:n−1|zn−1)p(zn|zn−1)p(zn−1) = p(xn|zn) zn−1 α(zn−1)A.
  • 32. 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(x1:n|zn)p(zn) = p(xn|zn)p(x1:n−1|zn)p(zn) = p(xn|zn)p(x1:n−1, zn) = p(xn|zn) zn−1 p(x1:n−1, zn|zn−1)p(zn−1) = p(xn|zn) zn−1 p(x1:n−1|zn−1)p(zn|zn−1)p(zn−1) = p(xn|zn) zn−1 α(zn−1)A.
  • 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) ?
  • 40. 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) = p(xn+1:N|zn) = zn+1 p(xn+1:N|zn, zn+1)p(zn+1|zn) = zn+1 p(xn+1:N|zn+1)p(zn+1|zn) = zn+1 p(xn+2:N|zn+1)p(xn+1|zn+1)p(zn+1|zn) = zn+1 β(zn+1)p(xn+1|zn+1)AT .
  • 41. 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) = p(xn+1:N|zn) = zn+1 p(xn+1:N|zn, zn+1)p(zn+1|zn) = zn+1 p(xn+1:N|zn+1)p(zn+1|zn) = zn+1 p(xn+2:N|zn+1)p(xn+1|zn+1)p(zn+1|zn) = zn+1 β(zn+1)p(xn+1|zn+1)AT .
  • 42. 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) = p(xn+1:N|zn) = zn+1 p(xn+1:N|zn, zn+1)p(zn+1|zn) = zn+1 p(xn+1:N|zn+1)p(zn+1|zn) = zn+1 p(xn+2:N|zn+1)p(xn+1|zn+1)p(zn+1|zn) = zn+1 β(zn+1)p(xn+1|zn+1)AT .
  • 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
  • 44. 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) = p(xn+1:N|zn) = zn+1 p(xn+1:N|zn, zn+1)p(zn+1|zn) = zn+1 p(xn+1:N|zn+1)p(zn+1|zn) = zn+1 p(xn+2:N|zn+1)p(xn+1|zn+1)p(zn+1|zn) = zn+1 β(zn+1)p(xn+1|zn+1)AT .
  • 45. 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) = p(xn+1:N|zn) = zn+1 p(xn+1:N|zn, zn+1)p(zn+1|zn) = zn+1 p(xn+1:N|zn+1)p(zn+1|zn) = zn+1 p(xn+2:N|zn+1)p(xn+1|zn+1)p(zn+1|zn) = zn+1 β(zn+1)p(xn+1|zn+1)AT .
  • 46. 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) = p(xn+1:N|zn) = zn+1 p(xn+1:N|zn, zn+1)p(zn+1|zn) = zn+1 p(xn+1:N|zn+1)p(zn+1|zn) = zn+1 p(xn+2:N|zn+1)p(xn+1|zn+1)p(zn+1|zn) = zn+1 β(zn+1)p(xn+1|zn+1)AT .
  • 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
  • 48. 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) = p(xn+1:N|zn) = zn+1 p(xn+1:N|zn, zn+1)p(zn+1|zn) = zn+1 p(xn+1:N|zn+1)p(zn+1|zn) = zn+1 p(xn+2:N|zn+1)p(xn+1|zn+1)p(zn+1|zn) = zn+1 β(zn+1)p(xn+1|zn+1)AT .
  • 49. 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) = p(xn+1:N|zn) = zn+1 p(xn+1:N|zn, zn+1)p(zn+1|zn) = zn+1 p(xn+1:N|zn+1)p(zn+1|zn) = zn+1 p(xn+2:N|zn+1)p(xn+1|zn+1)p(zn+1|zn) = zn+1 β(zn+1)p(xn+1|zn+1)AT .
  • 50. 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) = p(xn+1:N|zn) = zn+1 p(xn+1:N|zn, zn+1)p(zn+1|zn) = zn+1 p(xn+1:N|zn+1)p(zn+1|zn) = zn+1 p(xn+2:N|zn+1)p(xn+1|zn+1)p(zn+1|zn) = zn+1 β(zn+1)p(xn+1|zn+1)AT .
  • 51. 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) = p(xn+1:N|zn) = zn+1 p(xn+1:N|zn, zn+1)p(zn+1|zn) = zn+1 p(xn+1:N|zn+1)p(zn+1|zn) = zn+1 p(xn+2:N|zn+1)p(xn+1|zn+1)p(zn+1|zn) = zn+1 β(zn+1)p(xn+1|zn+1)AT .
  • 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).
  • 61. 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
  • 63. 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).
  • 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
  • 66. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
  • 67. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
  • 68. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
  • 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
  • 70. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
  • 71. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
  • 72. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
  • 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
  • 74. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
  • 75. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
  • 76. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
  • 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
  • 78. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
  • 79. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
  • 80. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ξ(zn−1, zn) = p(zn−1, zn|x1:N) = p(x1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1, zn)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn−1, zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) ...
  • 81. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn|zn−1)p(zn−1) p(x1:N) = α(zn−1)p(xn|zn)p(zn|zn−1)β(zn) p(x1:N) .
  • 82. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn|zn−1)p(zn−1) p(x1:N) = α(zn−1)p(xn|zn)p(zn|zn−1)β(zn) p(x1:N) .
  • 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
  • 84. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn|zn−1)p(zn−1) p(x1:N) = α(zn−1)p(xn|zn)p(zn|zn−1)β(zn) p(x1:N) .
  • 85. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn|zn−1)p(zn−1) p(x1:N) = α(zn−1)p(xn|zn)p(zn|zn−1)β(zn) p(x1:N) .
  • 86. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn|zn−1)p(zn−1) p(x1:N) = α(zn−1)p(xn|zn)p(zn|zn−1)β(zn) p(x1:N) . β(zn) α(zn)
  • 87. The forward-backward algorithm E-step of EM algorithm for HMM Evaluation of ξ(zn-1, zn) Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn−1, zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn−1, zn) p(x1:N) = p(x1:n−1|zn−1)p(xn|zn)p(xn+1:N|zn)p(zn|zn−1)p(zn−1) p(x1:N) = α(zn−1)p(xn|zn)p(zn|zn−1)β(zn) p(x1:N) .
  • 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).
  • 91. 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
  • 92. 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
  • 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
  • 101. 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
  • 102. 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
  • 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
  • 104. The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA p(xN+1|X) = zN+1 p(xN+1, zN+1|X) = zN+1 p(xN+1|zN+1, X)p(zN+1|X) = zN+1 p(xN+1|zN+1)p(zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN, zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) ...
  • 105. The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA p(xN+1|X) = zN+1 p(xN+1, zN+1|X) = zN+1 p(xN+1|zN+1, X)p(zN+1|X) = zN+1 p(xN+1|zN+1)p(zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN, zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) ...
  • 106. The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA p(xN+1|X) = zN+1 p(xN+1, zN+1|X) = zN+1 p(xN+1|zN+1, X)p(zN+1|X) = zN+1 p(xN+1|zN+1)p(zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN, zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) ...
  • 107. The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA zN zN+1z1 xN xN+1x1 Head Tail
  • 108. The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA p(xN+1|X) = zN+1 p(xN+1, zN+1|X) = zN+1 p(xN+1|zN+1, X)p(zN+1|X) = zN+1 p(xN+1|zN+1)p(zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN, zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) ...
  • 109. The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA p(xN+1|X) = zN+1 p(xN+1, zN+1|X) = zN+1 p(xN+1|zN+1, X)p(zN+1|X) = zN+1 p(xN+1|zN+1)p(zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN, zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) ...
  • 110. The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA p(xN+1|X) = zN+1 p(xN+1, zN+1|X) = zN+1 p(xN+1|zN+1, X)p(zN+1|X) = zN+1 p(xN+1|zN+1)p(zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN, zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) ...
  • 111. The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA p(xN+1|X) = zN+1 p(xN+1, zN+1|X) = zN+1 p(xN+1|zN+1, X)p(zN+1|X) = zN+1 p(xN+1|zN+1)p(zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN, zN+1|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) ...
  • 112. The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN) p(zN, X) p(X) = 1 p(X) zN+1 p(xN+1|zN+1) zN p(zN+1|zN)α(zN).
  • 113. The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN) p(zN, X) p(X) = 1 p(X) zN+1 p(xN+1|zN+1) zN p(zN+1|zN)α(zN).
  • 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
  • 115. The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN) p(zN, X) p(X) = 1 p(X) zN+1 p(xN+1|zN+1) zN p(zN+1|zN)α(zN).
  • 116. The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN) p(zN, X) p(X) = 1 p(X) zN+1 p(xN+1|zN+1) zN p(zN+1|zN)α(zN).
  • 117. The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN) p(zN, X) p(X) = 1 p(X) zN+1 p(xN+1|zN+1) zN p(zN+1|zN)α(zN).
  • 118. The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN) p(zN, X) p(X) = 1 p(X) zN+1 p(xN+1|zN+1) zN p(zN+1|zN)α(zN).
  • 119. The forward-backward algorithm Predictive distribution Aug 4, 2014 PRML 13.2.2 Shinichi TAMURA ... = zN+1 p(xN+1|zN+1) zN p(zN+1|zN, X)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN)p(zN|X) = zN+1 p(xN+1|zN+1) zN p(zN+1|zN) p(zN, X) p(X) = 1 p(X) zN+1 p(xN+1|zN+1) zN p(zN+1|zN)α(zN). Only depend on α(zN) i.e., whole X don't need
  • 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
  • 122. 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
  • 123. 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
  • 124. 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