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We see the problem below:

8.4 Multiple Hypothesis Testing. We see the problem below:. 1. Bayes mathod. Example 8.8: Multiple Hypothesis Testing probolm. Assuming v,a,b are independent of each other ， prior probability of two kind hypothesis are equal, find minimum error probability expression.

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We see the problem below:

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  1. 8.4 Multiple Hypothesis Testing We see the problem below: 1. Bayes mathod

  2. Example 8.8: Multiple Hypothesis Testing probolm Assuming v,a,b are independent of each other，prior probability of two kind hypothesis are equal, find minimum error probability expression.

  3. 2. Uniform most powerful testing(UMP) Example: level detection of gauss white noise environment. H1: zi=A+vi i=1,2,…,N H0: zi=vi i=1,2,…,N Vi‾N(0,2), A>0, find decision expression, and calculate decision performance.

  4. Decision threshold is independent of A, UMP exist

  5. When A<0, Realize UMP testing If sign of A is unknown Decision is relate to A, UMP does not exist.

  6. The sign of A is unknown, using two sided testing

  7. 8.4 Multiple Hypothesis Testing 3. broad sense likelihood ratio i=1,2,...,N Are ML estimations of 1，0 under H1 and H0 hypothesis. Using Neyman-Pearson criteria

  8. 8.4 Multiple Hypothesis Testing esample i=1,2,...N

  9. 8.4 Multiple Hypothesis Testing

  10. Threshold  is decided by false alarm ratio

  11. Example of detection - Constant Alarm False Ratio,CFAR example A and variance 2 is unknown。

  12. assume Pf is independent of noise

  13. 8.4 Multiple Hypothesis Testing assume

  14. Cal C1(z) Select minimun decision Cal C2(z) Cal CM-1(z)

  15. if MAP Criteria If P(H0)=P(H1) ML Criteria

  16. Example: multiple level detection of gauss white noise. min

  17. 0

  18. Example: Binary classification Black white Signal Processing Example-Pattern recognition Pattern recognition:wish to classify each pixel of image according to gray level. Could have M different gray levels.

  19. Actual gray level depends on lighting,orientation,etc. Model pixel values in a region as random with Where only mean level changes. To minimize Pe use MAP rule. Assume P(Hi)=1/Muse ML rule Maximum over i

  20.                          image data window pixel pixel to be classified To maximize  minimize Minimum distance receiver For binary classification,assume

  21. Let x=vector of all data(pixel) values in windows Choose whichever is smaller or decide H1 if We do this for all pixel assumes that all pixels within window have same mean value

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