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Unformatted text preview: Lecture 19 In the last lecture we found the Bayes decision rule that minimizes the Bayes eror = k X i =1 ( i ) i = k X i =1 ( i ) i ( 6 = H i ) . Let us write down this decision rule in the case of two simple hypothesis H 1 , H 2 . For simplicity of notations, given the sample X = ( X 1 , . . . , X n ) we will denote the joint p.d.f. by f i ( X ) = f i ( X 1 ) . . . f i ( X n ) . Then in the case of two simple hypotheses the Bayes decision rule that minimizes the Bayes error = (1) 1 ( 6 = H 1 ) + (2) 2 ( 6 = H 2 ) is given by = H 1 : (1) f 1 ( X ) > (2) f 2 ( X ) H 2 : (2) f 2 ( X ) > (1) f 1 ( X ) H 1 or H 2 : (1) f 1 ( X ) = (2) f 2 ( X ) or, equivalently, = H 1 : f 1 ( X ) f 2 ( X ) > (2) (1) H 2 : f 1 ( X ) f 2 ( X ) < (2) (1) H 1 or H 2 : f 1 ( X ) f 2 ( X ) = (2) (1) (19.1) (Here 1 = + , 1 = 0.) This kind of test if called likelihood ratio test since it is expressed in terms of the ratio f 1 ( X ) /f 2 ( X ) of likelihood functions. Example. Suppose we have only one observation X 1 and two simple hypotheses H 1 : = N (0 , 1) and H 2 : = N (1 , 1) . Let us take an apriori distribution given by (1) = 1 2 and (2) = 1 2 , 71 LECTURE 19. 72 i.e. both hypothesis have equal weight, and find a Bayes decision rulei....
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This note was uploaded on 10/11/2009 for the course STATISTICS 18.443 taught by Professor Dmitrypanchenko during the Spring '09 term at MIT.
 Spring '09
 DmitryPanchenko
 Statistics

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