电子科技大学：《信号检测与估计 Signal Detection and Estimation》课程教学资源（课件讲稿）Chapter 09、10 Bayesian Approach

Bayesian Approach Unknowns A is deterministic minimize MSE MSE(A(A-A)p(x:A)dx Unknowns is RV>minimize mean MSE Busa(④=4-)p(x,A)dxdA Use Bayesian rule p(x,A)=p(Ax)p(x) whxiong@uestc.edu.cn

whxiong@uestc.edu.cn Unknowns A is deterministic  minimize MSE Bayesian Approach Use Bayesian rule MSE(A b ) = Z (A b ¡ A) 2 p(x; A)dx Unknowns is RV minimize mean MSE BMSE(A b ) = Z Z (A ¡ A b ) 2 p(x; A)dxdA p(x;A) = p(Ajx)p(x)

Bayesian Approach Unknowns A is deterministic>minimize MSE MSE(A(A-A)p(x:A)dx Unknowns is RV->minimize mean MSE Busa(④=4-)p(x,A)dxdA Use Bayesian rule p(x,A)=p(Ax)p(x) wsa=∫/A- whxiong@uestc.edu.cn 17

whxiong@uestc.edu.cn Unknowns A is deterministic  minimize MSE Bayesian Approach 17 Use Bayesian rule MSE(A b ) = Z (A b ¡ A) 2 p(x; A)dx Unknowns is RV minimize mean MSE BMSE(A b ) = Z Z (A ¡ A b ) 2 p(x; A)dxdA p(x;A) = p(Ajx)p(x) BMSE(A b ) = Z ·Z (A ¡ A b ) 2 p(Ajx)dA ¸ p(x)dx

Bayesian Approach Unknowns A is deterministic minimize MSE MSE(A(A-A)p(x:A)dx Unknowns is RV->minimize mean MSE BAsE(A)=(A-A(A)dxdA Use Bayesian rule p(x,A)=p(Ax)p(x) Buss④=∫A-Ap4 dA)d& Define A=厂A-A2 p(Abe)dA whxiong@uestc.edu.cn 18

whxiong@uestc.edu.cn Unknowns A is deterministic  minimize MSE Bayesian Approach 18 Use Bayesian rule MSE(A b ) = Z (A b ¡ A) 2 p(x; A)dx Unknowns is RV minimize mean MSE BMSE(A b ) = Z Z (A ¡ A b ) 2 p(x; A)dxdA p(x;A) = p(Ajx)p(x) BMSE(A b ) = Z ·Z (A ¡ A b ) 2 p(Ajx)dA ¸ p(x)dx Define J(A b ) = Z (A ¡ A b ) 2 p(Ajx)dA

Bayesian Approach We want to minimize BMsE BMsE(④=JA,x刘pxdk whxiong@uestc.edu.cn 19

whxiong@uestc.edu.cn Bayesian Approach 19 We want to minimize BMSE BMSE(A b ) = Z J(A b ; x)p(x)dx

Bayesian Approach We want to minimize BMsE BM5E(④=J(x)p(x)d& If J is minimized for every x,then BMsE is minimized whxiong@uestc.edu.cn 20

whxiong@uestc.edu.cn Bayesian Approach 20 We want to minimize BMSE BMSE(A b ) = Z J(A b ; x)p(x)dx If J is minimized for every x, then BMSE is minimized