Similarly, P(H)=PHDP(H) (4.6) p(y) In order to calculate the a posteriori probabilities, it is necessary to know the a priori probabilities. Let P(Ho)=πo P(H)=π1 UESTC 6
Similarly, In order to calculate the a posteriori probabilities, it is necessary to know the a priori probabilities. Let 6 UESTC 1 1 1 ( | ) ( ) ( | ) (4.6) ( ) p y H P H P H y p y = 0 0 P(H ) = 1 1 P(H ) =
Do:the event associated with decision Ho D:the event associated with decision H D p(y|H1)π1p(y|Ho)πo (4.10) Do p(yHi) 元0 (4.11) p(y|Ho)0元1 Likelihood DI ratio 人L(y) TMAP UESTC DO 7
7 UESTC D0 : the event associated with decision H0 D1 : the event associated with decision H1 1 0 1 1 0 0 ( | ) ( | ) (4.10) D D p y H p y H 1 1 0 0 1 0 1 0 ( | ) (4.11) ( | ) ( ) D D D MAP D p y H p y H L y Likelihood ratio
4.2.1 Maximum-Likelihood case If two hypotheses are equally likely a priori, then 元=兀1= 2 Tmap 1 Maximum-likelihood (ML)criterion with a threshold Ev =1 It is often used in detection problems when there is no knowledge of the a priori probabilities. UESTC 8
8 UESTC 4.2.1 Maximum-Likelihood case • If two hypotheses are equally likely a priori, then 0 1 1 , 1 2 = = = MAP Maximum-likelihood (ML) criterion with a threshold It is often used in detection problems when there is no knowledge of the a priori probabilities. ML =1
Example 4.1 Consider the example where so =-b,s=b occurring with a priori probabilitieso=0.8 and元，=0.2.We will assume that b is larger than zero.If the noise pdf is a zero-mean Gaussian random variable with variance 902 (4.13) UESTC 9
9 UESTC Example 4.1 Consider the example where occurring with a priori probabilities and . We will assume that b is larger than zero. If the noise pdf is a zero-mean Gaussian random variable with variance , 0 1 s b s b = − = , 0 = 0.8 1 = 0.2 2 2 2 1 ( ) exp( ) (4.13) 2 2 n n p n = −
The two hypotheses conditional pdf lHy-" (4.14) (4.15) The threshold 元0 TMAP = 4 (4.16) UESTC π1 10
10 UESTC 2 1 2 1 ( ) ( | ) exp( ) (4.14) 2 2 n y b p y H − = − 2 0 2 1 ( ) ( | ) exp( ) (4.15) 2 2 n y b p y H + = − 0 1 MAP 4 (4.16) = = The two hypotheses conditional pdf The threshold