Probability estimation X x x T O T P(O|x,)=bb3a…bxo P(XIa)=t, axn,ar XT-1XT
11 Probability Estimation T oT P O X bx o bx o bx ( | , ) ... 1 1 2 2 = T T P X x ax x ax x ax x 1 1 2 2 3 1 ( | ) ... − = p o1 ot-1 ot ot+1 oT x1 xt-1 xt xt+1 xT
Probability estimation X x x T O T P(O|X,)=b0b20,…b P(XIu=Tax.ra T-I T P(O,XIu)=P(OIX, u)P(XIu 12
12 Probability Estimation P(O, X | ) = P(O | X,)P(X | ) T oT P O X bx o bx o bx ( | , ) ... 1 1 2 2 = T T P X x ax x ax x ax x 1 1 2 2 3 1 ( | ) ... − = p o1 ot-1 ot ot+1 oT x1 xt-1 xt xt+1 xT
Probability estimation X x x T O T P(OX,u=bb.b P(Xa=Tarr.a T-Ix P(O,XIu=POX,uP(XIu PO|)=∑POx,A)P(X|A)
13 Probability Estimation P(O, X | ) = P(O | X,)P(X | ) T oT P O X bx o bx o bx ( | , ) ... 1 1 2 2 = T T P X x ax x ax x ax x 1 1 2 2 3 1 ( | ) ... − = p = X P(O | ) P(O | X,)P(X | ) o1 ot-1 ot ot+1 oT x1 xt-1 xt xt+1 xT
Probability estimation X x x T O T PO)=∑zbn∏ab x1…
14 1 1 1 1 1 1 1 1 { ... } 1 ( | ) P + + + − = = t t t t T x x x o T x x t P O p x bx o a b Probability Estimation o1 ot-1 ot ot+1 oT x1 xt-1 xt xt+1 xT
Forward Procedure X x x T O T Special structure gives us an efficient solution using ynamic programming Intuition: Probability of the first t observations is the same for all possible t+1 length state sequences D efine. (1)=P(01O t 2t ilu 15
15 ( ) ( ... , | ) i t = P o1 ot xt = i Forward Procedure o1 ot-1 ot ot+1 oT x1 xt-1 xt xt+1 xT • Special structure gives us an efficient solution using dynamic programming. • Intuition: Probability of the first t observations is the same for all possible t+1 length state sequences. • Define: