Forward Procedure X x x T O T ∑ P( jP( ∑P(01O,x1=jx,=)P(x,=)P(On1x1=) 1.N ∑PO,x,=1)P(x1=jx,=1)P(+1x1= ∑a()a,b
21 = + + + = + + = + + + = + + = = = = = = = = = = = = = = = i N i i j j o t t t t i N t t t t t i N t t t t t i N t t t t t a b P o o x i P x j x i P o x j P o o x j x i P x i P o x j P o o x i x j P o x j 1... 1 1 1 1... 1 1 1 1... 1 1 1 1 1... 1 1 1 ( ) ( ... , ) ( | ) ( | ) ( ... , | ) ( ) ( | ) ( ... , , ) ( | ) o1 ot-1 ot ot+1 oT x1 xt-1 xt xt+1 xT Forward Procedure
Forward Procedure X x x T O T ∑P(1、Q,x=1,x1=)P( i=1..N ∑P(a1、O,x1=川x=1)P(x,=) P(O1+1 j ∑P(a )P(x+1=jx=)P(O+1 2
22 = + + + = + + = + + + = + + = = = = = = = = = = = = = = = i N i i j j o t t t t i N t t t t t i N t t t t t i N t t t t t a b P o o x i P x j x i P o x j P o o x j x i P x i P o x j P o o x i x j P o x j 1... 1 1 1 1... 1 1 1 1... 1 1 1 1 1... 1 1 1 ( ) ( ... , ) ( | ) ( | ) ( ... , | ) ( ) ( | ) ( ... , , ) ( | ) o1 ot-1 ot ot+1 oT x1 xt-1 xt xt+1 xT Forward Procedure
Forward Procedure X x x T O T ∑ P( jP(o+x1=j) ∑P(a1、O,x1=x=1)P(x1=)P(On1x1= 1.N ∑P P(x1=jx1=)P(On1|x1=) 1.N
23 = + + + = + + = + + + = + + = = = = = = = = = = = = = = = i N i i j j o t t t t i N t t t t t i N t t t t t i N t t t t t a b P o o x i P x j x i P o x j P o o x j x i P x i P o x j P o o x i x j P o x j 1... 1 1 1 1... 1 1 1 1... 1 1 1 1 1... 1 1 1 ( ) ( ... , ) ( | ) ( | ) ( ... , | ) ( ) ( | ) ( ... , , ) ( | ) o1 ot-1 ot ot+1 oT x1 xt-1 xt xt+1 xT Forward Procedure
Backward procedure X x x T O T B(T+1)=1 B,(0=P(O,01x=i Probability of the rest Iven the A(0)=∑ab(+)ls j=1.N 24
24 (t) P(o ...o | x i) i = t T t = o1 ot-1 ot ot+1 oT x1 xt-1 xt xt+1 xT Backward Procedure i (T +1) =1 = = + j N i i j i o j t a b t t 1... ( ) ( 1) Probability of the rest of the states given the first state
The solution to estimation X x x T O T P(O|)=∑a(T) Forward Procedure i=1 P(O|)=∑xB(1) Backward procedure P(OIu)=2a(),() Combination
25 o1 ot-1 ot ot+1 oT x1 xt-1 xt xt+1 xT The Solution to Estimation = = N i P O i T 1 ( | ) ( ) = = N i P O i i 1 ( | ) p (1) ( | ) ( ) ( ) 1 P O t t i N i i = = Forward Procedure Backward Procedure Combination