17HypothesisTesting

# 17HypothesisTesting - Hypothesis Testing Jeffrey B Burl EE...

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EE 3180 Probability and Random  Signal Analysis,  © 2009 by J. B. Burl 1 Hypothesis Testing Jeffrey B. Burl

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EE 3180 Probability and Random  Signal Analysis,  © 2009 by J. B. Burl 2 Binary Detection: The Source There are two hypotheses:  H 0  and  H 1 One of these hypotheses is true. We will assume the source is random  and generates hypotheses with the  probabilities: P ( H 0 ) P ( H 1 )
EE 3180 Probability and Random  Signal Analysis,  © 2009 by J. B. Burl 3 The Cost of Failure A decision rule can be either correct or  incorrect, leading to four possible outcomes: D 0  and  H 0  (Decide  H 0  and  H 0  true)   Correct D 1  and  H 0  (Decide  H 1  and  H 0  true)   Incorrect D 1  and  H 1  (Decide  H 1  and  H 1  true)   Correct D 0  and  H 1  (Decide  H 0  and  H 1  true)   Incorrect Assign a cost to every hypothesis-decision  combination: C 00 C 10 C 11 , and  C 01 C ik  is the cost incurred when we decide  H i  ( D i ) and  H k  is true.

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EE 3180 Probability and Random  Signal Analysis,  © 2009 by J. B. Burl 4 The Average Cost The expected value of the cost (the  average cost in the book) is 00 0 0 10 1 0 ( and ) ( and ) C C P D H C P D H = + 11 1 1 01 0 1 ( and ) ( and ) C P D H C P D H + +
EE 3180 Probability and Random  Signal Analysis,  © 2009 by J. B. Burl 5 The Average Cost, Cont. The probability that we decide  H 0  ( D 0 ) and  H 1   is true can be expanded in terms of the  source probability and the conditional  probability: The average cost can then be written: 0 1 1 0 1 ( and ) ( ) ( | ) P D H P H P D H = 00 0 0 0 10 0 1 0 ( ) ( | ) ( ) ( | ) C C P H P D H C P H P D H = + 11 1 1 1 01 1 0 1 ( ) ( | ) ( ) ( | ) C P H P D H C P H P D H + +

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EE 3180 Probability and Random  Signal Analysis,  © 2009 by J. B. Burl 6 The Average Cost, Cont. The conditional probability on the previous  slide is a function of the probabilistic  transition mechanism and the decision rule. We would like to separate these two effects to  simplify computation of this conditional  probability.
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