29 Assignment 3

Assignment 3 is be completed individually or with a partner (i.e., the maximum group size is two). Please submit the assignment as a single pdf or html file (one file per individual/group). Save the file as Yourlastname_Assignment3 (e.g., Hefley_Assignment3). Make sure to show your work in R to ensure that I can reproduce your results (e.g., figures, calculations, etc). Upload your completed assignment to Canvas before 5 pm on Tuesday 9/29/20.

29.1 Motivation

Assignment 3 is designed to give you experience with sampling-based approaches to fit Bayesian hierarchical models to spatio-temporal data.

29.2 Problems

29.3 Part 1

Suppose that we lived in a world where there was a test for COVID-19 that was 100% accurate (i.e., a test with no false-negatives and no false positives). In this world, a fictitious version of me decided to get tested for COVID-19 last week.

  1. Let the process model be \([y|\phi]=\text{Bernoulli}(\phi)\) and the parameter model be \([\phi]=\text{Beta}(\alpha,\beta)\). Use Bayes’ theorem to find \([\phi|y]\). Show all of your work. You can insert a image of your work if you want to use pencil and paper rather than typesetting it. Note that because the test is 100% accurate there is no need for a data model when using the Bayesian hierarchical modeling framework.

  2. Prior to being tested for COVID-19, I was unsure if I was infected. For the Bayesian model from #1, determine the exact parameter model (i.e., the values of the \(\alpha\) and \(\beta\)) that represent this prior knowledge.

  3. My test results came back negative. Using the data from my test results and your results from problem #1 and #2, obtain 1,000 samples from \([\phi|y]\). With these samples, represent \([\phi|y]\) using a histogram. Make sure to put appropriate axis labels on this histogram.

29.4 Part 2

For most Bayesian models, you won’t be able to find the posterior distribution (i.e., \([\phi|y]\)) using mathematical tools. Instead you will only be able to obtain samples using computational tools. Below you will use two different algorithms to obtain samples from \([\phi|y]\).

  1. Use your results from problem 1-3 and rejection sampling to obtain 1,000 samples from \([\phi|y]\). With these samples, represent \([\phi|y]\) using a histogram. Make sure to put appropriate axis labels on this histogram.

  2. Use your results from problem 1-3 and a Metropolis or Metropolis-Hastings algorithm to obtain 1,000 samples from \([\phi|y]\). With these samples, represent \([\phi|y]\) using a histogram. Make sure to put appropriate axis labels on this histogram.

29.5 Part 3

For most tests, there is a probability of obtaining a false-negative test result. Because there is the potential for false-negative test results, we need to include a data model in our Bayesian hierarchical model. To accommodate the potential for false-negatives use the data model \[[z|y]=\begin{cases} \text{Bernoulli(0.9)} & \text{if}\:y=1\\ 0 & \text{if}\:y=0 \end{cases}\] where \(z\) is my test result (which was negative), 0.9 is the probability that a true positive (i.e., \(y=1\)) result in a positive test result (i.e., \(z=1\)).

  1. Obtain samples from the posterior distribution \([y,\phi|z]\). With these samples, represent \([y|z]\) using one histogram and \([\phi|z]\) with another histogram. Make sure to put appropriate axis labels on this histogram.

  2. Given my test results, what is the probability I have COVID-19?