ECE313.Lecture17

# ECE313.Lecture17 - ECE 313 Probability with Engineering...

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Decision-making under uncertainty III Professor Dilip V. Sarwate Department of Electrical and Computer Engineering © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign. All Rights Reserved ECE 313 Probability with Engineering Applications

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ECE 313 - Lecture 17 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 45 Hypothesis testing model One of M ≥ 2 mutually exclusive hypotheses H 0 , H 1 , … , H M–1 is true X is a random variable whose value we can observe, and use, to decide which of the hypotheses is true If H i happens to be the true hypothesis, then the pmf of X is P i (u) We can think of P i (u) as p X | H i (u | H i ), the conditional pmf of X given that H i is true
ECE 313 - Lecture 17 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 45 The decision rule We observe the value of X and announce our decision as to which hypothesis we believe to be true This decision may or may not coincide with reality — our decision may be H i when in fact H j is the true hypothesis The decision rule (which we are free to choose as we wish) assigns a hypothesis ( the decision! ) to each possible value of X

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ECE 313 - Lecture 17 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 45 The likelihood matrix L Likelihood matrix L has M rows (one for each hypothesis) and N columns (one for each value taken on by X ) The entry in the i-th row and j-th column is P i (u j ), the (conditional) probability that X = u j when H i is the true hypothesis The sum of the entries in each row is 1 The decision rule is speciFed by shading one entry in each column of L
ECE 313 - Lecture 17 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 45 (Conditional) error probabilities The (conditional) probability of a correct decision given that H i is true is the sum of the shaded squares on the i-th row The (conditional) probability of error given that H i is true is the sum of the unshaded squares on the i-th row Many statisticians are very uncomfortable with calling these parameters conditional probabilities

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ECE 313 - Lecture 17 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 45 Maximum-likelihood decision rule The hypothesis for which the probability of the observation is the maximum is called the maximum-likelihood (ML) decision when that observation is made For each observation (value of X ), the ML decision rule maximizes the likelihood of the observation Operationally, the ML rule says: shade the largest entry in each column of L
ECE 313 - Lecture 17 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 45 Probabilities of error for ML rule The ML decision rule is just one of many possible decision rules, and the various

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## This note was uploaded on 09/29/2009 for the course ECE 123 taught by Professor Mr.pil during the Spring '09 term at University of Iowa.

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ECE313.Lecture17 - ECE 313 Probability with Engineering...

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