GE 331-Lecture 21

GE 331-Lecture 21 - Likelihood Approach to Hypothesis...

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IE 300/GE 331 Lecture 21 Negar Kiyavash, UIUC 1 Likelihood Approach to Hypothesis Testing: Set-up: Two possible hypotheses H 0 and H 1 Likelihood of observing a certain ranges of X under each hypothesis: p X (x; H 0 ) and p X (x; H 1 ) Observation X (in general it is a vector). To make our life easy in this course we only deal with scalar observations. Decision rule: Find a decision rule that maps the realized value of observation x to one of the hypothesis In our treatment of BHT in this class: The decision rule divides the observation space to two subspaces corresponding to 1) Acceptance region of H 0 (also rejection region of H 1 ); 2) Rejection region of H 0 (also acceptance region of H 1 ) Acceptance Region R c Rejection Region R
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IE 300/GE 331 Lecture 21 Negar Kiyavash, UIUC 2 BHT and Associated Errors: Spaces of Possible Observations X H 0 true No error H 1 true No error H 0 true False rejection (false alarm) H 1 true False acceptance (miss) Rejection Region R for Null Hypothesis Acceptance Region R c for Null Hypothesis
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IE 300/GE 331 Lecture 21 Negar Kiyavash, UIUC 3 Likelihood Approach to BHT (cont.): • How do we choose the decision boundary? • Choose the decision boundary according to the likelihood ratio: L(x)=p X (x; H 1 ) /p X (x; H 0 ) • Note: we use the notation p X (x; H 1 ) in general because X can be discrete or continuous. • Rejection region R={x|L(x)> τ } • Specifically if learnt to decision rules: ML and MAP. •F o r M L : τ =1 and for MAP τ = p(H 0 ) /p(H 1 ) Acceptance Region R c Rejection Region R
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IE 300/GE 331 Lecture 21 Negar Kiyavash, UIUC 4 Likelihood Approach to BHT (cont.): • Recall that in the Bayesian framework the hypothesis are random variables. • Let us denote the random variable corresponding to the hypothesis by Θ .
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GE 331-Lecture 21 - Likelihood Approach to Hypothesis...

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