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R代写-STAT 8150/7150-Assignment2

时间：2021-04-06

Bayesian Data Analysis Assignment2: Due Week 8, 2021

STAT 8150/7150

Instructions:

This assignment covers weeks 4, 5, and 6. This assignment is worth 40 marks.

1. Due on 30th April 2021

2. For all the questions please provide the relevant mathematical derivations, the com-

puter programs (only using R software) and the plots.

3. Please submit on iLearn a single PDF file containing all your work (code, compu-

tations, plots, etc.). Other file formats (e.g. Word, html) will NOT be accepted.

4. Try to use Rmarkdown through Rstudio. But it is not compulsory to use Rmark-

down even if facilitate to reproduce results. Only upload the pdf file.

1 of 3

Bayesian Data Analysis Assignment2: Due Week 8, 2021

1. Question 1 (6 marks)

You are designing a very small experiment to determine the sensitivity of a new security

alarm system. You will simulate five robbery attempts and record the number of these

attempts that trigger the alarm. Because the dataset will be small you ask two experts

for their opinion. One expects the alarm probability to be 0.95 with standard deviation

0.05, the other expects 0.80 with standard deviation 0.20.

(a) (3 marks) Translate these two priors into beta PDFs, plot the two beta PDFs and

the corresponding mixture of experts prior with equal weight given to each expert.

(b) (3 marks) Now you conduct the experiment and the alarm is triggered in every

simulated robbery. Plot the posterior of the alarm probability under a uniform prior,

each experts’ prior, and the mixture of experts prior.

2. Question 2 (14 marks)

The data in the table below are the result of a survey of commuters in 10 counties likely

to be affected by a proposed addition of a high occupancy vehicle (HOV) lane.

County Approve Disapprove County Approve Disapprove

1 12 50 6 15 8

2 90 150 7 67 56

3 80 63 8 22 19

4 5 10 9 56 63

5 63 63 10 33 19

(a) (4 marks) Analyze the data in each county separately using the Jeffreys’ prior

distribution and report the posterior 95% credible set for each county.

(b) (3 marks) Let pˆi be the sample proportion of commuters in county i that approve of

the HOV lane (e.g., pˆ1 = 12/(12 + 50) = 0.194). Select a and b so that the mean and

variance of the Beta(a, b) distribution match the mean and variance of the sample

proportions pˆ1, . . . , pˆ10.

(c) (4 marks) Conduct an empirical Bayesian analysis by computing the 95% posterior

credible sets that results from analyzing each county separately using the Beta(a, b)

prior you computed in (b). (Reminder: empirical Bayesian analysis means you have

used the observed data to define your prior).

(d) (3 marks) Present he credible sets in the same plot for each county obtained from

(a) and (c). How do the results from (a) and (c) differ?

3. Question 3 (12 marks)

In this question we revisit the Question 2 from your first assignment. The NBA free throw

data are the same as before:

2 of 3

Bayesian Data Analysis Assignment2: Due Week 8, 2021

Overall Clutch Clutch

Player proportion makes attempts

Russell Westbrook 0.845 64 75

James Harden 0.847 72 95

Kawhi Leonard 0.880 55 63

LeBron James 0.674 27 39

Isaiah Thomas 0.909 75 83

Stephen Curry 0.898 24 26

Giannis Antetokounmpo 0.770 28 41

John Wall 0.801 66 82

Anthony Davis 0.802 40 54

Kevin Durant 0.875 13 16

We will fit the following model:

Yi | θi ∼ Binomial(ni, θi) and θi | m ∼ Beta[emqi, em(1− qi)],

where Yi is the number of made clutch shots for player i = 1, . . . , 10, ni is the number of

attempted clutch shots, qi ∈ (0, 1) is the overall proportion, and m ∼ Normal(0, 10).

(a) (2 marks) Explain why this is a reasonable prior for θi.

(b) (2 marks) Explain the role of m in the prior.

(c) (3 marks) Derive the conditional posterior for θ1: pi(θ1 | m, θ2, . . . , θ10, Y1, . . . , Y10).

(d) (3 marks) Fit your model in JAGS and comment your code.

(e) (2 marks) Present the 95% credible interval for all parameters of your model.

4. Question 4 (8 marks)

Suppose there are 10 power plant pumps. The number of failures of those 10 pumps follows

a Poisson distribution i.e. Xi ∼ Poisson(θiti), i = 1, 2, . . . , 10 where θi is the failure rate

for pump i and ti is the length of operation time of the pump (measured in 1,000s of

hours). The dataset is summarised into the following table:

Pump 1 2 3 4 5 6 7 8 9 10

ti 94.5 15.7 62.9 126 5.24 31.4 1.05 1.05 2.1 10.5

xi 5 1 5 14 3 19 1 1 4 22

A conjugate gamma prior distribution is adopted for the failure rates:

θi ∼ Γ(a, b), i = 1, 2, . . . , 10,

with a and b also have their prior distributions as

a ∼ Exponential(1) and b ∼ Γ(0.1, 1).

(a) (2 marks) Draw the DAG (Directed Acyclic Graph) corresponding to this model.

(b) (4 marks) Implement the above model in JAGS (comment your code).

(c) (2 marks) Is there any pump that appears to be broken down more often than the

others?

3 of 3

学霸联盟

STAT 8150/7150

Instructions:

This assignment covers weeks 4, 5, and 6. This assignment is worth 40 marks.

1. Due on 30th April 2021

2. For all the questions please provide the relevant mathematical derivations, the com-

puter programs (only using R software) and the plots.

3. Please submit on iLearn a single PDF file containing all your work (code, compu-

tations, plots, etc.). Other file formats (e.g. Word, html) will NOT be accepted.

4. Try to use Rmarkdown through Rstudio. But it is not compulsory to use Rmark-

down even if facilitate to reproduce results. Only upload the pdf file.

1 of 3

Bayesian Data Analysis Assignment2: Due Week 8, 2021

1. Question 1 (6 marks)

You are designing a very small experiment to determine the sensitivity of a new security

alarm system. You will simulate five robbery attempts and record the number of these

attempts that trigger the alarm. Because the dataset will be small you ask two experts

for their opinion. One expects the alarm probability to be 0.95 with standard deviation

0.05, the other expects 0.80 with standard deviation 0.20.

(a) (3 marks) Translate these two priors into beta PDFs, plot the two beta PDFs and

the corresponding mixture of experts prior with equal weight given to each expert.

(b) (3 marks) Now you conduct the experiment and the alarm is triggered in every

simulated robbery. Plot the posterior of the alarm probability under a uniform prior,

each experts’ prior, and the mixture of experts prior.

2. Question 2 (14 marks)

The data in the table below are the result of a survey of commuters in 10 counties likely

to be affected by a proposed addition of a high occupancy vehicle (HOV) lane.

County Approve Disapprove County Approve Disapprove

1 12 50 6 15 8

2 90 150 7 67 56

3 80 63 8 22 19

4 5 10 9 56 63

5 63 63 10 33 19

(a) (4 marks) Analyze the data in each county separately using the Jeffreys’ prior

distribution and report the posterior 95% credible set for each county.

(b) (3 marks) Let pˆi be the sample proportion of commuters in county i that approve of

the HOV lane (e.g., pˆ1 = 12/(12 + 50) = 0.194). Select a and b so that the mean and

variance of the Beta(a, b) distribution match the mean and variance of the sample

proportions pˆ1, . . . , pˆ10.

(c) (4 marks) Conduct an empirical Bayesian analysis by computing the 95% posterior

credible sets that results from analyzing each county separately using the Beta(a, b)

prior you computed in (b). (Reminder: empirical Bayesian analysis means you have

used the observed data to define your prior).

(d) (3 marks) Present he credible sets in the same plot for each county obtained from

(a) and (c). How do the results from (a) and (c) differ?

3. Question 3 (12 marks)

In this question we revisit the Question 2 from your first assignment. The NBA free throw

data are the same as before:

2 of 3

Bayesian Data Analysis Assignment2: Due Week 8, 2021

Overall Clutch Clutch

Player proportion makes attempts

Russell Westbrook 0.845 64 75

James Harden 0.847 72 95

Kawhi Leonard 0.880 55 63

LeBron James 0.674 27 39

Isaiah Thomas 0.909 75 83

Stephen Curry 0.898 24 26

Giannis Antetokounmpo 0.770 28 41

John Wall 0.801 66 82

Anthony Davis 0.802 40 54

Kevin Durant 0.875 13 16

We will fit the following model:

Yi | θi ∼ Binomial(ni, θi) and θi | m ∼ Beta[emqi, em(1− qi)],

where Yi is the number of made clutch shots for player i = 1, . . . , 10, ni is the number of

attempted clutch shots, qi ∈ (0, 1) is the overall proportion, and m ∼ Normal(0, 10).

(a) (2 marks) Explain why this is a reasonable prior for θi.

(b) (2 marks) Explain the role of m in the prior.

(c) (3 marks) Derive the conditional posterior for θ1: pi(θ1 | m, θ2, . . . , θ10, Y1, . . . , Y10).

(d) (3 marks) Fit your model in JAGS and comment your code.

(e) (2 marks) Present the 95% credible interval for all parameters of your model.

4. Question 4 (8 marks)

Suppose there are 10 power plant pumps. The number of failures of those 10 pumps follows

a Poisson distribution i.e. Xi ∼ Poisson(θiti), i = 1, 2, . . . , 10 where θi is the failure rate

for pump i and ti is the length of operation time of the pump (measured in 1,000s of

hours). The dataset is summarised into the following table:

Pump 1 2 3 4 5 6 7 8 9 10

ti 94.5 15.7 62.9 126 5.24 31.4 1.05 1.05 2.1 10.5

xi 5 1 5 14 3 19 1 1 4 22

A conjugate gamma prior distribution is adopted for the failure rates:

θi ∼ Γ(a, b), i = 1, 2, . . . , 10,

with a and b also have their prior distributions as

a ∼ Exponential(1) and b ∼ Γ(0.1, 1).

(a) (2 marks) Draw the DAG (Directed Acyclic Graph) corresponding to this model.

(b) (4 marks) Implement the above model in JAGS (comment your code).

(c) (2 marks) Is there any pump that appears to be broken down more often than the

others?

3 of 3

学霸联盟