Some detection criterions Chapter 4 Single Sample Detection of Binary Hypotheses UESTC 1
1 UESTC Some detection criterions Chapter 4 Single Sample Detection of Binary Hypotheses
4.1 Chapter highlights Hypothesis testing and the MaP criterion. Extension to the general Bayes criterion. ·inimax criterion Neyman-Person criterion. UESTC 2
2 UESTC 4.1 Chapter highlights • Hypothesis testing and the MAP criterion. • Extension to the general Bayes criterion. • Minimax criterion • Neyman-Person criterion
4.2 Hypothesis testing MAP criterion Consider two possible values so and s.We wish to decide between these two alternatives when we observe a measurement corrupted by noise. y=S,+n,i=0,1 (4.1) Ho:the value so being present H:the value s being present The a priori probabilities: UESTC P(Ho)=πo P(H)=π1 3
3 UESTC 4.2 Hypothesis testing & MAP criterion Consider two possible values s0 and s1 . We wish to decide between these two alternatives when we observe a measurement corrupted by noise. H0 : the value s0 being present H1 : the value s1 being present , 0,1 (4.1) i y s n i = + = 0 0 P(H ) = 1 1 P(H ) = The a priori probabilities:
MAP criterion (Maximum A Posteriori):If P(H y)>P(H y) (4.2) we decide so are present,and conversely we decide S if P(H y)>P(H y) (4.3) Using Bayes'theorem,we can write P(Hy)= p(yH)P(H) (4.4) p(y) UESTC 4
4 UESTC 0 1 P H y P H y ( | ) ( | ) (4.2) 1 0 P H y P H y ( | ) ( | ) (4.3) 0 0 0 ( | ) ( ) ( | ) (4.4) ( ) p y H P H P H y p y MAP criterion (Maximum A Posteriori) :If we decide s0 are present, and conversely we decide s1 if Using Bayes’ theorem, we can write
P(H y)P(y)=P(y H)P(H) P(H。Iy)p(y)(y)=p(y|H。)(dy)P(H。) Likelihood function: y=S+n,i=0,1 (4.1) Conditional PDF (Likelihood function: p(yH) p(ylH。) UESTC 5
5 UESTC , 0,1 (4.1) i y s n i = + = 0 p y H ( | ) 1 p y H ( | ) Likelihood function: Conditional PDF (Likelihood function:) 0 0 0 0 0 0 ( | ) ( ) ( | ) ( ) ( | ) ( )( ) ( | )( ) ( ) P H y P y P y H P H P H y p y dy p y H dy P H