Description

General information
• The recommended tool in this course is R (with the IDE R-Studio). You can download R here and R-Studio here. There are tons of tutorials, videos and introductions to R and R-Studio online. You can find some initial hints here.
• You can write the report with your preferred software, but the outline of the report should follow the instruction in the R markdown template that can be found here.
• Report all results in a single, anonymous *.pdf -file and return it to peergrade.io.
• The course has its own R package with data and functionality to simplify coding. To install the package just run the following:
1. install.packages(“remotes”)
2. remotes::install_github(“avehtari/BDA_course_Aalto”, subdir = “rpackage”)
• Many of the exercises can be checked automatically using the R package markmyassignment. Information on how to install and use the package can be found here. • Additional self study exercises and solutions for each chapter in BDA3 can be found here.
• We collect common questions regarding installation and technical problems in a course Frequently Asked Questions (FAQ). This can be found here. • If you have any suggestions or improvements to the course material, please feel free to create an issue or submit a pull request to the public repository!!
Information on this assignment
This exercise is related to Chapters 2 and 3. The maximum amount of points from this assignment is 9. Use Frank Harrell’s recommendations on how to state results in Bayesian two group comparisons (and note that there is no point null hypothesis testing in this exercises).
Reading instructions: Chapter 2 and 3 in BDA3, see reading instructions here and here.
Grading instructions: The grading will be done in peergrade. All grading questions and evaluations for assignment 3 can be found here.
To use markmyassignment for this assignment, run the following code in R:
> library(markmyassignment) > exercise_path <-
“https://github.com/avehtari/BDA_course_Aalto/blob/master/exercises/tests/ex3.yml”
> set_assignment(exercise_path)
> # To check your code/functions, just run
> mark_my_assignment()

1. Inference for normal mean and deviation (3 points)
A factory has a production line for manufacturing car windshields. A sample of windshields has been taken for testing hardness. The observed hardness values y1 can be found in file windshieldy1.txt. The data can also be accessed from the aaltobda R package as follows:
> library(aaltobda)
> data(“windshieldy1”)
> head(windshieldy1)
[1] 13.357 14.928 14.896 15.297 14.820 12.067
Below are test examples that can be used. The functions below can also be tested with markmyassignment. Note! This is only a test case. You need to change to the full data windshieldy above when reporting your results.
> windshieldy_test <- c(13.357, 14.928, 14.896, 14.820)
In the report, formulate (1) model likelihood, (2) the prior, and (3) the resulting posterior.
a) What can you say about the unknown µ? Summarize your results using Bayesian point estimate (i.e. E(µ|y)), a posterior interval (95%), and plot the density. A test example can be found below for an uninformative prior. Note! Posterior intervals are also called credible intervals and are different from confidence intervals.
> mu_point_est(data = windshieldy_test)
[1] 14.5
> mu_interval(data = windshieldy_test, prob = 0.95)
[1] 13.3 15.7
b) What can you say about the hardness of the next windshield coming from the production line before actually measuring the hardness? Summarize your results using Bayesian point estimate, a predictive interval (95%), and plot the density. A test example can be found below.
> mu_pred_point_est(data = windshieldy_test)
[1] 14.5
> mu_pred_interval(data = windshieldy_test, prob = 0.95)
[1] 11.8 17.2
Note! Predictive intervals are different from posterior intervals.
Hint With a conjugate prior a closed form posterior is Student’s t form (see equations in the book). R users can use the dt function after doing input normalisation. We have added an R function dtnew() in the aaltobda R package which does that. For generating samples, you can use the corresponding rtnew function.
2. Inference for the difference between proportions (3 points)
In the report, formulate (1) model likelihood, (2) the prior, and (3) the resulting posterior.
a) Summarize the posterior distribution for the odds ratio, (p1/(1 − p1))/(p0/(1 − p0)). Compute the point estimate, a posterior interval (95%), and plot the histogram. Use Frank Harrell’s recommendations how to state results in Bayesian two group comparison. Below is a test case on how the odd ratio should be computed. Note! This is only a test case. You need to change to the real posteriors when reporting your results.
> set.seed(4711)
> p0 <- rbeta(100000, 5, 95)
> p1 <- rbeta(100000, 10, 90)
> posterior_odds_ratio_point_est(p0 = p0, p1 = p1)
[1] 2.676
> posterior_odds_ratio_interval(p0 = p0, p1 = p1, prob = 0.9)
[1] 0.875 6.059
b) Discuss the sensitivity of your inference to your choice of prior density with a couple of sentences.
Hint With a conjugate prior, a closed-form posterior is the Beta form for each group separately (see equations in the book). You can use rbeta() to sample from the posterior distributions of p0 and p1, and use these samples and odds ratio equation to get samples from the distribution of the odds ratio.
3. Inference for the difference between normal means (3 points)
Consider a case where the same factory has two production lines for manufacturing car windshields. Independent samples from the two production lines were tested for hardness. The hardness measurements for the two samples y1 and y2 are given in the files windshieldy1.txt and windshieldy2.txt. These can be accessed directly with
> data(“windshieldy1”)
> data(“windshieldy2”)
We assume that the samples have unknown standard deviations σ1 and σ2.
In the report, formulate (1) model likelihood, (2) the prior, and (3) the resulting posterior. Use uninformative or weakly informative priors and answer the following questions:
a) What can you say about µd = µ1 − µ2? Summarize your results using a Bayesian point estimate, a posterior interval (95%), and plot the histogram. Use Frank Harrell’s recommendations how to state results in Bayesian two group comparison.
b) What is the probability that the means are the same? Explain your reasoning with a couple of sentences.
Hint With a conjugate prior, a closed-form posterior is Student’s t form for each group separately (see equations in the book). You can use rt() function to sample from the posterior distributions of µ1 and µ2, and use these samples to get samples from the distribution of the difference µd = µ1 −µ2. The equivalent function in R is the rt function. Be careful to scale them and shift them according to their mean and variance values in R, as described above.
Hint Posterior distributions of µ1 and µ2 are continuous, and thus the posterior distribution of the difference µd = µ1 − µ2 is also continuous. What is the probability that µd = 0?