BSDA Assignment 2 Solved

Description

 

The recommended tool in this course is R (with the IDE R-Studio). You can download R here and R-Studio here. There are many tutorials, videos and introductions to R and R-Studio online. You can nd some initial hints from RStudio Education pages.

When working with R, we recommend writing the report using R markdown and the provided R markdown template. The template includes the formatting instructions and how to include code and gures.

Instead of R markdown, you can use other software to make the PDF report, but you should use the same instructions for formatting. These instructions are also available in the PDF produced from the R markdown template.

We supply a Google Colab notebook that you can also use for the assignments. We have included the installation of all necessary R packages; hence, this can be an alternative to using your own local computer. You can nd the notebook here. You can also open the notebook in Colab here.

Report all results in a single and anonymous pdf. Note that no other formats are allowed.

The course has its own R package bsda with data and functionality to simplify coding. To install the package, just run the following (upgrade=”never” skips question about updating other packages):

1. packages(“remotes”)
2. remotes::install_github(“MansMeg/BSDA”, subdir = “rpackage”, upgrade=”never”)

Many of the exercises can be checked automatically using the R package markmyassignment. you can           nd information on how to install and use the package here. There is no need to include markmyassignment results in the report.

You can nd common questions and answers regarding the installation and technical problems in Frequently Asked Questions (FAQ).

You can nd deadlines and information on how to turn in the assignments in Studium.

You are allowed to discuss assignments with your friends, but it is not permitted to copy solutions directly from other students or the internet. Try to solve the actual assignment problems with your code and explanations. Do not share your answers publicly. We compare the answers with the “urkund” system. We will report all suspected plagiarism.

If you have any suggestions or improvements to the course material, please post in the course chat feedback channel, create an issue, or submit a pull request to the public repository here.

It is mandatory to include the following parts in all assignments (these are included already in the template):

1. Time used for reading: How long time took the reading assignment (in hours)
2. Time used for the assignment: How long time took the basic assignment (in hours)
3. Good with assignment: Write one-two sentences of what you liked with the assignment/what we should keep for next year.
4. Things to improve in the assignment: Write one-two sentences of what you think can be improved in the assignment. Can something be clari ed further? Did you get stuck on stu unrelated to the content of the assignment etc.

You can nd information on how each assignment will be graded and how points are assigned here. Note! This grading information can change during the course, for example, if we nd errors or inconsistencies. Please feel free to comment on these grading instructions, ideally before turning in your assignment, if you think something is missing or is incorrect.

To pass (G) the assignment, you need 70% of the total points. To pass with distinction (VG), you need 90% of the total points. See the grading information on the point allocations for each assignment.

Information on this assignment

This assignment is related to Chapters 1 and 2.

Reading instructions: Chapter 1 and 2 in BDA3, see reading instructions. You may nd an additional discussion about choosing priors by Andrew Gelman useful, they can be found here.

To use markmyassignment for this assignment, run the following code in R:

library(markmyassignment) assignment_path <paste(“https://github.com/MansMeg/BSDA/”, “blob/main/assignments/tests/assignment2.yml”, sep=””) set_assignment(assignment_path) # To check your code/functions, just run mark_my_assignment()

Don’t include markmyassignment results in the report.

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 the algae dataset (’0’: no algae, ’1’: algae present). The data can be accessed from the bsda R package as follows:

library(bsda) data(“algae”) head(algae) ## [1] 0 1 1 0 0 0 # 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 model for the observations y and a Beta(2,10) prior for binomial model parameter 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 and it su ces to refer to that derivation. Also, it is not necessary to write out the distributions; it is su cient to use label-parameter format, e.g. Beta(·,·).

Your task is to make Bayesian inference for binomial model and answer questions based on it:

1. formulate (1) the likelihood p(y|π) as a function of 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.
2. 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 di erent from con dence intervals. Note! In your report, use the values from the data algae, not algae_test.

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

1. 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

1. What assumptions are required in order to use this kind of a model with this type of data? (No need to discuss exchangeability yet, as it is discussed in more detail in BDA Chapter 5 and Lecture 7)
2. Make prior sensitivity analysis by testing a couple of di erent reasonable priors and plot the di erent 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.