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Assignment 2 Problem - Bayesian Inference – 2007 | STAT 664, Assignments of Statistics

George Mason University (GMU)Statistics
Prof. Kathryn Laskey

Material Type: Assignment; Professor: Laskey; Class: Bayesian Inference/Dec Theory; Subject: Statistics; University: George Mason University; Term: Unknown 1989;

Typology: Assignments

Pre 2010

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SYST/STAT 664: Homework Assignment 2
due February 5, 2007
Homework is due at class time on the date indicated. You may submit on paper or electronically
via Blackboard. Please make sure your name is on every page of the assignment, and it is clearly
marked which question you are answering. Your response will be graded for correctness and
clarity.
1. Problem 3 of Assignment 1 is concerned with the decision of whether to administer sanctions
to an individual who may have considered a security violation. We have the following losses:
• Do not administer sanction; individual is innocent loss = 0
• Administer sanction; individual is guilty loss = 1
• Administer sanction; individual is innocent loss = 10
• Do not administer sanction; individual is guilty loss = 100
We can obtain evidence that has 80% sensitivity and 85% specificity.
a. Find the range of prior probabilities for which considering the evidence results in
lower expected loss than ignoring or not collecting the evidence.
b. There is a loss associated with collecting the evidence (dollars, time spent, emotional
distress). As a function of the prior probability p, find the maximum loss for which
running the test could be justified.
2. In an experiment, subjects were given the choice between two gambles:
Gamble 1:
A: $2500 with probability 0.33 B: $2400 with certainty
$2400 with probability 0.66
$0 with probability 0.01
Suppose that a person is an expected utility maximizer. Set the utility scale so that u($0) = 0
and u($2500) = 1. Denote u($2400) by x. For what values of x would a person choose
Option A? For what values would a person choose Option B?
Gamble 2:
C: $2500 with probability 0.33 D: $2400 with probability 0.34
$0 with probability 0.67 $0 with probability 0.66
For what values of x would a person choose Option C? For what values would a person
choose Option D? Explain why no expected utility maximizer would prefer B in Gamble 1
and C in Gamble 2.
This problem is a version of the famous Allais paradox, named after the prominent critic of
subjective expected utility theory who first presented it. Many people’s choices violate
pf2

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SYST/STAT 664: Homework Assignment 2

due February 5, 2007 Homework is due at class time on the date indicated. You may submit on paper or electronically via Blackboard. Please make sure your name is on every page of the assignment, and it is clearly marked which question you are answering. Your response will be graded for correctness and clarity.

  1. Problem 3 of Assignment 1 is concerned with the decision of whether to administer sanctions to an individual who may have considered a security violation. We have the following losses:
    • Do not administer sanction; individual is innocent loss = 0
    • Administer sanction; individual is guilty loss = 1
    • Administer sanction; individual is innocent loss = 10
    • Do not administer sanction; individual is guilty loss = 100 We can obtain evidence that has 80% sensitivity and 85% specificity. a. Find the range of prior probabilities for which considering the evidence results in lower expected loss than ignoring or not collecting the evidence. b. There is a loss associated with collecting the evidence (dollars, time spent, emotional distress). As a function of the prior probability p , find the maximum loss for which running the test could be justified.
  2. In an experiment, subjects were given the choice between two gambles: Gamble 1: A: $2500 with probability 0.33 B: $2400 with certainty $2400 with probability 0. $0 with probability 0. Suppose that a person is an expected utility maximizer. Set the utility scale so that u($0) = 0 and u($2500) = 1. Denote u($2400) by x. For what values of x would a person choose Option A? For what values would a person choose Option B? Gamble 2: C: $2500 with probability 0.33 D: $2400 with probability 0. $0 with probability 0.67 $0 with probability 0. For what values of x would a person choose Option C? For what values would a person choose Option D? Explain why no expected utility maximizer would prefer B in Gamble 1 and C in Gamble 2. This problem is a version of the famous Allais paradox , named after the prominent critic of subjective expected utility theory who first presented it. Many people’s choices violate

subjective expected utility theory. For example, Kahneman and Tversky^1 found that 82% of subjects preferred B over A, and 83% preferred C over D.

  1. Consider the following gamble. Gamble 3: This is a two-stage problem. In the first stage there is a 66% chance you will win nothing and quit. There is a 34% chance you will go on to the second stage. In the second stage you may choose between the following two options. E: $2500 with probability 33/34 F: $2400 with certainty $0 with probability 1/ Verify that the probability distributions of the outcomes for E and C are the same. Verify also that the probability distributions of the outcomes for F and D are the same. Ask five people what their choices would be in Gambles 1, 2, and 3. Make sure the people you ask have not already provided answers to this question for someone else in SYST/STAT
    1. Record the three choices each person made in a table, where the rows of the table are people and the columns are preferences for Gambles 1, 2, and 3. If a person chooses the most common pattern of B, C, and F, explain the equivalence of D and F, and observe whether the explanation changes their preference. If any of your subjects changed their responses, add a fourth, fifth and sixth column of your table and record the new results for the subject(s) who changed. (^1) Kahneman, D., P. Slovic, et al. (1982). Judgment under Uncertainty: Heuristics and Biases. Cambridge, Cambridge University Press.