Bayesian Statistics Statistics 4224/5224 — Spring 2021
Take-home final exam
Do any two or all three of the following problems, and submit your answers to Courseworks by the end of the day on Thursday, April 22.
The exam is open-book and open-note, but “closed-internet,” and any collaboration among classmates is strictly prohibited. That is, you may consult course materials (textbook, lecture notes, homework solutions) as much as you wish, but conducting a Google-search for hints would absolutely, unambigu- ously, be considered cheating. You may not discuss the exam questions with anyone other than the instructor. Any suspected violation of these rules will be referred to the Office of Student Conduct and Community Standards.
1. Chapter 9 Exercise 2 from Gelman et al.
Oscar has lost his dog; there is a 70% probability it is in forest A and a 30% chance it is in forest B. If the dog is in forest A and Oscar looks there for a day, he has a 50% chance of finding the dog. If the dog is in forest B and Oscar looks there for a day, he has an 80% chance of finding the dog.
(a) If Oscar can search only one forest for a day, where should he look to maximize his probability of finding the dog? Justify your answer.
(b) Assume Oscar made the rational decision and the dog is still lost (and is still in the same forest as yesterday). Where should he search for the dog on the second day? Justify your answer.
(c) Again assume Oscar made the rational decision on the second day and the dog is still lost (and still in the same forest). Where should he search on the third day? Justify your answer.
(d) Suppose Oscar will search for at most three days, with the following payoffs: −1 if the dog is found in one day, −2 if the dog is found on the second day, −3 if the dog is found on the third day, and −10 otherwise.
i. What is Oscar’s expected payoff?
ii. What is Oscar’s expected payoff if he knows the dog is in forest A?
iii. What is Oscar’s expected payoff if he knows the dog is in forest B?
iv. Before the search begins, how much should Oscar be willing to pay to be told which forest his dog is in?
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2. (Based on Chapter 14 Exercise 1 from Gelman et al.)
The following table gives short-term radon measurements for a sample of houses in three counties in Minnesota.
County Blue Earth
Clay Goodhue
Radon measurements (pCi/L)
5.0, 13.0, 7.2, 6.8, 12.8, 5.8∗, 9.5, 6.0, 3.8, 14.3∗, 1.8, 6.9, 4.7, 9.5
0.9∗, 12.9, 2.6, 3.5∗, 26.6, 1.5, 13.0, 8.8, 19.5, 2.5∗, 9.0, 13.1, 3.6, 6.9∗
14.3, 6.9∗, 7.6, 9.8∗, 2.6, 43.5, 4.9, 3.5, 4.8, 5.6, 3.5, 3.9, 6.7
All measurements were recorded on the basement level of the houses, except for those indicated with asterisks, which were recorded on the first floor.
Fit a Bayesian linear regression to the logarithms of the radon measurements in the above table, with indicator variables for the three counties and for whether a measurement was recorded on the first floor.
(a) Report the posterior median and 50% and 95% posterior intervals for the mean log radon concentration at each of the six house types (three counties; measurement taken in basement or on first floor). Prepare a summary table and comment on your findings.
(b) Plot the posterior predictive density for the log radon concentration in a single house, for each of the six house types. Put all six graphs in a single display (a 2 × 3 array of plots), using a consistent scale for horizontal and vertical axes (so plots are directly comparable). Comment on your results.
(c) Give a 95% prediction interval for the radon concentration in a single house, on the original (unlogged) scale, for each of the six house types.
3. Consider the NBA free throw data first introduced in Homework 1 Problem 6.
Player
Russell Westbrook James Harden
Kawhi Leonard
LeBron James
Isaiah Thomas
Stephen Curry
Giannis Antetokounmpo John Wall
Anthony Davis
Kevin Durant
Overall proportion
Clutch makes
Clutch attempts
0.845 64 75 0.847 72 95 0.880 55 63 0.674 27 39 0.909 75 83 0.898 24 26 0.770 28 41
0.801 66 82
0.802 40 54
0.875 13 16
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Assume that for player i,
yi|θ ∼ indep Binomial(ni, θi) ,
where yi is the number of clutch makes, ni is the number of clutch attempts, and θi is the clutch
make probability. Fit the model
logit(θi) = α + βxi ,
where xi = logit(qi) is the log odds of making a regular free throw (qi is the overall free throw
percentage). Assume the noninformative prior p(α, β) ∝ 1.
(a) Summarize the posterior distributions of α and β.
(b) Use posterior predictive checks to verify the model fits well. Use the test statistics
T1(y)=meanof {y1/n1,…,yk/nk} T2(y)=SDof {y1/n1,…,yk/nk} T3(y) = max {y1/n1, . . . , yk/nk} T4(y)=min{y1/n1,…,yk/nk} .
(c) Who does the model predict is most likely to be the best clutch free throw shooter among the ten players, and with what probability? Who actually was the best for this season? Who does the model predict will be the worst clutch shooter (and with what probability), and who actually was the worst?
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