Problem Set 7: Decision-Making Under Uncertainty for ECE 313 at University of Illinois | Assignments Statistics

University of Illinois Fall 2009

ECE 313: Problem Set 7

Decision-Making Under Uncertainty

Due: Wednesday October 14 at 4 p.m.

Reading: Ross, Chapter 3; Powerpoint Lecture Slides, Sets 15-18

Decision-Making and Decision Making Under Uncertainty: Additional reading

material available on the COMPASS web page for ECE 313

Noncredit Exercises: Chapter 3: Problems 53, 58, 59, 62, 63, 70-74, 78, 81

Theoretical Exercises 6, 7(a), 25, 26; Self-Test Problems 15-26.

Reminder: Hour Exam I on Monday October 12, 7:00 p.m. – 8:00 p.m., 100 Noyes Lab.

Coverage of material is Problem Sets 1-6

1. [A warm-up exercise]

If H0is the true hypothesis, the random variable Xtakes on values 0, 1, 2, and 3 with probabilities

0.1, 0.2, 0.3, and 0.4 respectively. If H1is the true hypothesis, the random variable Xtakes on values

0, 1, 2, and 3 with probabilities 0.4, 0.3, 0.2, and 0.1 respectively.

(a) Find the likelihood matrix Land indicate the maximum-likelihood decision rule by shading the

appropriate entries in L. What is the false-alarm probability PFA and what is the missed-detection

probability PMD for the maximum-likelihood decision rule?

(b) Now suppose that the hypotheses have a priori probabilities π0= 0.7 and π1= 0.3. Use the law

of total probability to find the average error probability of the maximum-likelihood decision rule

that you found in part (a).

on it the Bayesian decision rule, which is also known as the minimum-error-probability (MEP)

or maximum a posteriori probability (MAP) decision rule. What is the average error probability

of the Bayesian decision rule? Is it smaller or larger than the average error probability of the

maximum-likelihood decision rule? In the latter case, provide a brief explanation as to why the

minimum-error-probability rule has a larger average error probability than another rule.

(d) Show that in each of the two cases i= 0 and for i= 1, it is true that if πi>0.8, then the

Bayesian decision rule always decides that Hiis the true hypothesis, no matter what the value of

Xis. Hint: Remember that π1−i= 1 −πi.

2. [Which route did she fly?]

We return to Problem 6 of Problem Set 6 and consider Bob who is observing Alice’s arrival in New

York. Let H0denote the hypothesis that Alice flew via Chicago and H1the hypothesis that Alice flew

via Denver. Let Xdenote the number of hours that Alice is late in arriving in New York.

(a) Given that H0is the true hypothesis, what is the pmf of X? Given that H1is the true hypothesis,

what is the pmf of X? Hint: you may have found the answers already when you worked on

Problem Set 6, but if not, feel free to use the results in the posted Solutions to Problem Set 6.

Use these pmfs to construct the likelihood matrix L.

(b) The maximum-likelihood decision rule that Bob uses in deciding which route Alice flew depends

on the unspecified values of pand q. Assume that 0 <p<1 and 0 < q < 1, and remember that

qis not necessarily equal to 1 −p. Consider the columns of Lone by one, and for each column,

determine for what values of pand qthe maximum-likelihood decision is in favor of H0and for

what (hopefully other) values of pand qthe maximum-likelihood decision is in favor of H1.

regardless of the values of pand q. Hint: PFA = sum of unshaded entries on H0row, etc.

(d) Repeat part (b) for the Bob’s Bayesian decision rule assuming that π0=3

4and π1=1

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University of Illinois Fall 2009

ECE 313: Problem Set 7

Decision-Making Under Uncertainty

Due: Wednesday October 14 at 4 p.m. Reading: Ross, Chapter 3; Powerpoint Lecture Slides, Sets 15- Decision-Making and Decision Making Under Uncertainty: Additional reading material available on the COMPASS web page for ECE 313 Noncredit Exercises: Chapter 3: Problems 53, 58, 59, 62, 63, 70-74, 78, 81 Theoretical Exercises 6, 7(a), 25, 26; Self-Test Problems 15-26. Reminder: Hour Exam I on Monday October 12, 7:00 p.m. – 8:00 p.m., 100 Noyes Lab. Coverage of material is Problem Sets 1-

[A warm-up exercise] If H 0 is the true hypothesis, the random variable X takes on values 0, 1, 2, and 3 with probabilities 0.1, 0.2, 0.3, and 0.4 respectively. If H 1 is the true hypothesis, the random variable X takes on values 0, 1, 2, and 3 with probabilities 0.4, 0.3, 0.2, and 0.1 respectively.

(a) Find the likelihood matrix L and indicate the maximum-likelihood decision rule by shading the appropriate entries in L. What is the false-alarm probability PFA and what is the missed-detection probability PMD for the maximum-likelihood decision rule? (b) Now suppose that the hypotheses have a priori probabilities π 0 = 0.7 and π 1 = 0.3. Use the law of total probability to find the average error probability of the maximum-likelihood decision rule that you found in part (a). (c) Use the a priori probabilities given in part (b) to find the joint probability matrix J and indicate on it the Bayesian decision rule, which is also known as the minimum-error-probability (MEP) or maximum a posteriori probability (MAP) decision rule. What is the average error probability of the Bayesian decision rule? Is it smaller or larger than the average error probability of the maximum-likelihood decision rule? In the latter case, provide a brief explanation as to why the minimum-error-probability rule has a larger average error probability than another rule. (d) Show that in each of the two cases i = 0 and for i = 1, it is true that if πi > 0 .8, then the Bayesian decision rule always decides that Hi is the true hypothesis, no matter what the value of X is. Hint: Remember that π 1 −i = 1 − πi.

[Which route did she fly?] We return to Problem 6 of Problem Set 6 and consider Bob who is observing Alice’s arrival in New York. Let H 0 denote the hypothesis that Alice flew via Chicago and H 1 the hypothesis that Alice flew via Denver. Let X denote the number of hours that Alice is late in arriving in New York.

(a) Given that H 0 is the true hypothesis, what is the pmf of X? Given that H 1 is the true hypothesis, what is the pmf of X? Hint: you may have found the answers already when you worked on Problem Set 6, but if not, feel free to use the results in the posted Solutions to Problem Set 6. Use these pmfs to construct the likelihood matrix L. (b) The maximum-likelihood decision rule that Bob uses in deciding which route Alice flew depends on the unspecified values of p and q. Assume that 0 < p < 1 and 0 < q < 1, and remember that q is not necessarily equal to 1 − p. Consider the columns of L one by one, and for each column, determine for what values of p and q the maximum-likelihood decision is in favor of H 0 and for what (hopefully other) values of p and q the maximum-likelihood decision is in favor of H 1. (c) Explain why for the maximum-likelihood decision rule, PFA ≤ 1 − 2 p(1 − p) and PMD ≤ 1 − q^2 regardless of the values of p and q. Hint: PFA = sum of unshaded entries on H 0 row, etc. (d) Repeat part (b) for the Bob’s Bayesian decision rule assuming that π 0 = 34 and π 1 = 14.

[Detection problem with Poisson distributed observations] A certain deep space transmitter uses on-off modulation of a laser to send a bit. The result is that: if a ZERO is sent, the number of photons X arriving at the receiver has the Poisson distribution with mean λ 0 = 2; and if a ONE is sent, X has the Poisson distribution with mean λ 1 = 6.

(a) Describe the ML decision rule. Express it as directly in terms of X as possible. (b) Describe the MAP decision rule assuming that sending a ZERO is a priori five times more likely than sending a ONE (i.e. π 0 /π 1 = 5). Express the rule as directly in terms of X as possible.

[Detection problem with a geometric model] A transmitter chooses one of two routes (Route 0 or Route 1) and repeatedly transmits a packet over the chosen route until the packet is received without error (that is, without CRC checksum failure) at the receiver. X denotes the number of times the packet is transmitted over the chosen route including the final error-free transmission. Assuming that the successive transmissions are independent trials of an experiment, the two hypotheses are - H 1 : Route 1 is used for packet transmission: X ∼ Geometric(p 1 ) - H 0 : Route 0 is used for packet transmission: X ∼ Geometric(p 0 )

where 0 < p 1 < p 0 < 1 are the probabilities of error-free transmission over the two routes.

(a) State the maximum-likelihood decision rule as to which route was used as a threshold test on the observed value of X. (b) Suppose the transmitter chooses Route 0 and Route 1 with probabilities π 0 and π 1 = 1 − π 0 respectively, i.e., π 0 and π 1 are the a priori probabilities of hypotheses H 0 and H 1. Assume that 0 < π 0 < 1. For what values of π 0 (if any) does the minimum-error-probability decision rule always choose hypothesis H 1 regardless of the value of the observation X? For what values of π 0 (if any) does the minimum-error-probability decision rule always choose hypothesis H 0 regardless of the value of the observation X?

[“Give me an F!” shouted the cheerleader... ] H 0 , H 1 , and H 2 respectively denote the hypotheses that a student is excellent, good, or average (there are no poor students). The number of grade points earned by the student in a course is a random variable X that takes on values 3, 6, 9, and 12 only. The professor knows that the pmf of X when H 0 is true is p 0 (12) = 0. 75 , p 0 (9) = 0. 15 , p 0 (6) = 0. 08 , p 0 (3) = 0.02, that is, an excellent student has 75% chance of doing well enough on the exam to get an A, 15% chance of a B, etc. Similarly, when H 1 is the true hypothesis, the pmf of X is p 1 (12) = 0. 15 , p 1 (9) = 0. 6 , p 1 (6) = 0. 15 , p 1 (3) = 0.1, while if H 2 is true, p 2 (12) = 0. 05 , p 2 (9) = 0. 1 , p 2 (6) = 0. 65 , p 2 (3) = 0.2. The professor observes X and must decide which of the hypotheses H 0 , H 1 , H 2 is true.

(a) What is the professor’s maximum-likelihood decision rule? (b) What is the probability that an excellent student is mistakenly labeled as good? What is the probability that an excellent student is mistakenly labeled as average? What is the probability that an average student is classified either as good or as excellent? (c) If P (H 0 ) = 0. 2 , P (H 1 ) = 0.55, and P (H 2 ) = 0.25, what is the probability that the maximum- likelihood decision rule mis-classifies students? (d) What is the Bayes’ decision rule corresponding to these probabilities and what is the probability that the Bayes’ decision rule mis-classifies students? (e) At the Lake Wobegon campus of the University, 95% of students are excellent and 5% are good (and thus they are all above average!) What is Bayes’ decision rule in this case? That is, what does the Bayesian professor decide about a student based on the four possible results of the student’s exam?

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ECE 313: Problem Set 7

Decision-Making Under Uncertainty