> Risk with respect to @2 r2=n2i p(xo2)dx+m22p(xo2)dx Probabilities of wrong decisions weighted by the penalty terms Average risk r=F;P(1)+r2P(a2)
16 ➢Risk with respect to ➢ ➢Average risk 2 r p x d x p x d x R R ( ) ( ) 2 2 1 2 2 2 2 1 2 = + ( ) ( ) 1 1 2 P 2 r = rP + r Probabilities of wrong decisions, weighted by the penalty terms
.o Choose r and r so that r is minimized ☆ Then assign x to a,if 4=mP(ro,)P(o,)+miP(a2)P(02)< (2=12p(xa)Pa)+2p(xa2)P(a) ☆ Equivalently assign x in @@2)if p(xO)、/P(a)421-2 12 (x2)P(Oa)42-1 12: likelihood ratio
17 ❖ Choose and so that r is minimized ❖ Then assign to if ❖ Equivalently: assign x in if : likelihood ratio R1 R2 x i ( ) 1 2 1 2 1 1 2 1 2 2 1 2 2 1 1 2 ( ) ( ) ( ) ( ) ( ) − − P P p x p x 12 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 1 1 2 2 2 2 1 1 1 1 1 2 1 2 2 p x P p x P p x P p x P + +
今fP(0)=P(02)=1ann1=2=0 x>0, if P(a,)>P(xo2)21 x)02 if P(xo2)>P(xo)2 if n21=M2= Minimum classifica tion error probabilit y
18 error probabilit y if Minimum classifica tion if ( ) ( ) if ( ) ( ) and 0 2 1 ( ) ( ) 2 1 1 2 2 1 1 2 2 2 1 1 2 2 1 1 1 2 1 2 1 1 2 2 = → → = = = = x P x P x x P x P x P P ❖ If
☆ An example: P(ra) zAp(r2) p(x02)=-=ex(-(x-1) P(O1)=P(O2)= 00.5 L 1.00
19 ❖ An example: − = − = = − = − − − = − 1.0 0 0 0.5 2 1 ( ) ( ) exp( ( 1) ) 1 ( ) exp( ) 1 ( ) 1 2 2 2 2 1 L P P p x x p x x
Then the threshold value is X for minimum xo: exp(-x)=exp((x-1))= Threshold x for minimum r co: exp(x)=2exp((x-1))= (1-Cm2)
20 ➢Then the threshold value is: ➢Threshold for minimum r 2 1 : exp( ) exp( ( 1) ) for minimum : 0 2 2 0 0 = − = − − x x x x x Pe 2 1 2 (1 2) ˆ ˆ : exp( ) 2exp( ( 1) ) 0 2 2 0 − = − = − − n x x x x 0 x ˆ