# Assignment 4

# 1 General information

This assignment is related to Lecture 4 and Chapters 3 and 10.

**The maximum amount of points from this assignment is 6.**

We have prepared a **quarto template specific for this assignment (html, qmd, pdf)** to help you get started.

**Reading instructions:**

**Grading instructions:**

The grading will be done in peergrade. All grading questions and evaluations for this assignment are contained within this document in the collapsible **Rubric** blocks.

**For posterior statistics of interest, only report digits that are not completely random based on the Monte Carlo standard error (MCSE).**

*Example:* If you estimate \(E(\mu) \approx 1.234\) with MCSE(\(E(\mu)\)) = 0.01, then the true expectation is likely to be between \(1.204\) and \(1.264\), it makes sense to report \(E(\mu) \approx 1.2\).

See Lecture video 4.1, the chapter notes, and a case study for more information.

- The recommended tool in this course is R (with the IDE RStudio).
- Instead of installing R and RStudio on you own computer, see
**how to use R and RStudio remotely**. - If you want to install R and RStudio locally, download R and RStudio.
- There are tons of tutorials, videos and introductions to R and RStudio online. You can find some initial hints from
**RStudio Education pages**. - When working with R, we recommend writing the report using
`quarto`

and the provided template. The template includes the formatting instructions and how to include code and figures. - Instead of
`quarto`

, you can use other software to make the PDF report, but the the same instructions for formatting should be used. - Report all results in a single,
**anonymous***.pdf -file and submit it in**peergrade.io**. - The course has its own R package
`aaltobda`

with data and functionality to simplify coding. The package is pre-installed in JupyterHub. To install the package on your own system, run the following code (upgrade="never" skips question about updating other packages):

- Many of the exercises can be checked automatically using the R package
`markmyassignment`

(pre-installed in JupyterHub). Information on how to install and use the package can be found in the`markmyassignment`

documentation. There is no need to include`markmyassignment`

results in the report. - Recommended additional self study exercises for each chapter in BDA3 are listed in the course web page. These will help to gain deeper understanding of the topic.
- Common questions and answers regarding installation and technical problems can be found in Frequently Asked Questions (FAQ).
- Deadlines for all assignments can be found on the course web page and in Peergrade. You can set email alerts for the deadlines in Peergrade settings.
- You are allowed to discuss assignments with your friends, but it is not allowed to copy solutions directly from other students or from internet.
- You can copy, e.g., plotting code from the course demos, but really try to solve the actual assignment problems with your own code and explanations.
- Do not share your answers publicly.
- Do not copy answers from the internet or from previous years. We compare the answers to the answers from previous years and to the answers from other students this year.
- Use of AI is allowed on the course, but the most of the work needs to by the student, and you need to report whether you used AI and in which way you used them (See points 5 and 6 in Aalto guidelines for use of AI in teaching).
- All suspected plagiarism will be reported and investigated. See more about the
**Aalto University Code of Academic Integrity and Handling Violations Thereof**. - Do not submit empty PDFs, almost empty PDFs, copy of the questions, nonsense generated by yourself or AI, as these are just harming the other students as they can’t do peergrading for the empty or nonsense submissions. Violations of this rule will be reported and investigated in the same way was 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!

# 2 Bioassay model

In this exercise, you will use a dose-response relation model that is used in BDA3 Section 3.7 and in **the chapter reading instructions**. The used likelihood is the same, but instead of uniform priors, we will use a bivariate normal distribution as the joint prior distribution of the parameters \(\alpha\) and \(\beta\).

In the prior distribution for \((\alpha,\beta)\), the marginal distributions are \(\alpha \sim N(0,2^2)\) and \(\beta \sim N(10,10^2)\), and the correlation between them is \(\mathrm{corr}(\alpha, \beta)=0.6\).

The mean and covariance of the bivariate normal distribution are a length–\(2\) vector and a \(2 \times 2\) matrix. The elements of the covariance matrix can be computed using the relation of correlation and covariance.

You are given 4000 independent draws from the posterior distribution of the model in the dataset `bioassay_posterior`

in the `aaltobda`

package.

The answer is graded as correct only if the number of digits reported is correct. The number of significant digits can be different for the mean and quantile estimates. In some other cases, the number of digits reported can be less than MCSE allows for practical reasons as discussed in the lecture.

**Hint**:

Quantiles can be computed with the `quantile`

function. With \(S\) draws, the MCSE for \(\text{E}[\theta]\) is \(\sqrt{\text{Var} [\theta]/S}\). MCSE for the quantile estimates can be computed with the `mcse_quantile`

function from the `aaltobda`

package.

# 3 Importance sampling

Now we discard our posterior draws and switch to importance sampling.

Non-log importance ratios are given by equation (10.3) in the course book. The fact that our proposal distribution is the same as the prior distribution makes this task easier. The **logarithm** of the likelihood can be computed with the `bioassaylp`

function from the `aaltobda`

package. The data required for the likelihood can be loaded with `data("bioassay")`

.

Use the function `rmvnorm`

from the `aaltobda`

package for sampling.

Equation (10.4) in the course book.

*BDA3 1st (2013) and 2nd (2014) printing have an error for \(\tilde{w}(\theta^s)\) used in the effective sample size equation (10.4). The normalized weights equation should not have the multiplier S (the normalized weights should sum to one). The later printings, the online version, and the slides have the correct equation.*

The values below are *only* a test case, you need to use 4000 draws for \(\alpha\) and \(\beta\) in the final report.

Use the same equation for the MCSE of \(\text{E}[\theta]\) as earlier (\(\sqrt{\text{Var} [\theta]/S}\)), but now replace \(S\) with \(S_{\text{eff}}\). To compute \(\text{Var} [\theta]\) with importance sampling, use the identity \(\text{Var}[\theta] = \text{E}[\theta^2] - \text{E}[\theta]^2\).