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
Reading instructions: Chapter 1 and 2 in BDA3, see reading instructions here and here.
Grading instructions: The grading will be done in peergrade. All grading questions and evaluations for assignment 2 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/ex2.yml”
> set_assignment(exercise_path)
> # To check your code/functions, just run
> mark_my_assignment()

Inference for binomial proportion (Computer)
Algae status is monitored in 274 sites at Finnish lakes and rivers. The observations for the 2008 algae status at each site are presented in file algae.txt (’0’: no algae, ’1’: algae present). The data can also be accessed from the aaltobda R package as follows:
> library(aaltobda)
> data(“algae”)
> # the data is now stored in the variable ‘algae’
So that you can test the correctness of your code implementations, we provide some results for the following test data. It is also possible to check the functions you need to implement with markmyassignment.
> algae_test <- c(0, 1, 1, 0, 0, 0)
Note! This data is only for the tests, you need to change to the full data algae when reporting your results.
Let π be the probability of a monitoring site having detectable blue-green algae levels and y the observations in algae. Use a binomial likelihood for the observations y and a Beta(2,10) prior for π to formulate a Bayesian model. Here it is not necessary to derive the posterior distribution for π as it has already been done in the book. Also, it is not necessary to write out the distributions; it is sufficient to use label-parameter format, e.g. Beta(·,·).
Your task is to formulate a Bayesian model and answer questions based on it:
a) formulate (1) model likelihood p(y|π), (2) the prior p(π), and (3) the resulting posterior p(π|y). Report the posterior in the format Beta(·,·), where you replace ·’s with the correct numerical values.
b) What can you say about the value of the unknown π according to the observations and your prior knowledge? Summarize your results with a point estimate (i.e. E(π|y)) and a 90% posterior interval. Note! Posterior intervals are also called credible intervals and are different from confidence intervals.
> beta_point_est(prior_alpha = 2, prior_beta = 10, data = algae_test)
[1] 0.2222222
> beta_interval(prior_alpha = 2, prior_beta = 10, data = algae_test, prob = 0.9)
[1] 0.0846451 0.3956414
Note! Report the values using the data algae, not algae_test.
c) What is the probability that the proportion of monitoring sites with detectable algae levels π is smaller than π0 = 0.2 that is known from historical records?
> beta_low(prior_alpha = 2, prior_beta = 10, data = algae_test, pi_0 = 0.2)
[1] 0.4511238
d) What assumptions are required in order to use this kind of a model with this type of data?
e) Make prior sensitivity analysis by testing a couple of different reasonable priors and plot the different posteriors. Summarize the results by one or two sentences.
Hint! With a conjugate prior, a closed-form posterior is Beta form (see equations in the book). Useful functions: dbeta, pbeta, qbeta in R.