connecting S=s'ES and S=sES. Using Bayes'rule,the aposteriori probability can be expressed as P(u =ily)=P(u =i,S=s',S=sly) =∑pw,=4S=sS=s,y (se成 p(y) where B is the set of transitions S that are caused by the input us=i.Thus (4.8) can be written as p(u =1.S=s'.S.=s.y )p(y) L(u)=In n三u=0=5S=7/p0 (4.10) The sequence y in p(u=i,S=s'.S=s.y)can be written as p(u.()y.yx))=p(uS-Syiye-yi) where y=(yy)represents the portion of the received sequence y in the time interval to / Applying Bayes'rule to this expression yields the following decomposition of p(u=i,S-=s',S:=s,y): p(u=1.S-=5'.S=5.y)=p(u =1.S.=5.S-=5'.yi.y.y) =p(yim lyeyiu=i.S=5.S-=s)p(yyu=i.S:=s.S=s) =p(yiu ls=s)p(y.yf,u=i.S=s,S=s) (4.1) p(yiIs=s)p(u=i.S.=5.yS=5'.y)p(S-=s'.y) =p(S=s',y）p(4=iS=sylS=s）pylS=s) =a4-(s')(s',sB(s) (4.12) where (s)=p(S=s.y),called the forward state metric,is the joint of the encoder state at time k being S=s and receiving the sequencey B (s)=p(yS=s)is the backward state metric;and 419 4-19 connecting 1 ' k S s    and k S s   . Using Bayes’ rule, the a posteriori probability can be expressed as 1 11 ( ', ) ( | ) ( , ', | ) i k N N k kk k ss B Pu i Pu iS s S s        y y  1 1 ( ', ) 1 ( , ', , ) i ( ) k N kk k N ss B pu iS s S s p        y y where i Bk is the set of transitions S S k k 1  that are caused by the input uk=i. Thus (4.8) can be written as 1 0 1 11 ( ', ) 1 11 ( ', ) ( 1, ', , ) / ( ) ( ) ln ( 0, ', , ) / ( ) k k N N kk k ss B k N N kk k ss B pu S s S s p L u pu S s S s p              y y y y (4.10) The sequence y in 1 1 ( , ', , ) N kk k pu iS s S s    y can be written as     1 1 1 1 1 11 1 , , ,( ,., ), ,( ,., ) , , , , , k N kk k k k k N kk k kk pu S S pu S S      y y y y y y yy  where 1 ( , ,., ) l t tt l   y yy y represents the portion of the received sequence y in the time interval t to l. Applying Bayes’ rule to this expression yields the following decomposition of 1 1 ( , ', , ) N kk k pu iS s S s    y :    1 1 1 11 1 , ', , , , ', , , N kN k k k k k k kk pu iS s S s pu iS sS s          y y yy     1 1 11 1 1 1 |, , , , , , , , ' Nk k kk k k k k k k k p u iS sS s p u iS sS s            y yy yy    1 11 1 | , , , ' N k kk k k k k p S s p u iS sS s         y yy (4.11)      1 1 1 11 11 | , , | ', ', N kk k k k k kk k p S s pu iS s S s p S s            y yy y       1 11 1 1 ', , , | ' | k N k k k kk k k p S s pu iS s S s p S s           y yy 1( ') ( ', ) ( ) i kk k   s ss s   (4.12) where  k k k () ( , ) s pS s   y1 , called the forward state metric, is the joint of the encoder state at time k being Sk=s and receiving the sequence 1 k y ; k k N k () ( | ) sp Ss   y 1 is the backward state metric; and