ECE313.Lecture16

ECE313.Lecture16 - ECE 313 Probability with Engineering...

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Decision-making under uncertainty II 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 16 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 48 Hypothesis testing model One of M 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) To avoid trivialities, we assume that M ≥ 2
ECE 313 - Lecture 16 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 48 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 16 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 48 Specifying the decision rule We can specify the decision rule as a table that lists all the values of X and the corresponding decisions After observing the value of X , we merely look up our decision in the table and announce it! The process is completely mechanical (or computerized?) — observe X and look up the decision from the table
ECE 313 - Lecture 16 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 48 General remarks on decision rules There are M choices of hypothesis for the decision for each of the N values of X Thus, there are M N different decision rules that we could use If M > N, some hypotheses will never be chosen for any value of X If M < N, several values of X will result in the same decision

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ECE 313 - Lecture 16 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 48 Simple example of a decision rule Consider a binary hypothesis test in which the observation X takes on only the values 1 and 2 Our decision rule is as follows If X = 1, decide H 0 is the true hypothesis If X = 2, decide H 1 is the true hypothesis Remember that the decision might not be correct (or it might be correct)
ECE 313 - Lecture 16 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 48 Deterministic vs randomized rules The decision rule that we have described is called a deterministic decision rule A randomized decision rule is one in which, after observing X , we choose the decision “randomly” with specified probabilities If X = 1, toss a coin with P(Heads) = 0.7 and decide H 0 if Heads, H 1 if Tails If X = 2, toss a coin with P(Heads) = 0.25 and decide H if Heads, H if Tails

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