# 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 \(\mathrm{corr}(\alpha, \beta)=0.6\).

Report the mean (vector of two values) and covariance (two by two matrix) of the bivariate normal distribution.

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.

Report

- the mean as well as
- 5 \(\%\) and 95 \(\%\) quantiles separately

for both

- \(\alpha\) and
- \(\beta\).

Report also the Monte Carlo standard errors (MCSEs) for the mean and quantile estimates and explain in text what does Monte Carlo standard error mean and how you decided the number of digits to show.

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.

**Q3: **
Are the mean and covariance of the prior in a) reported? The correct answers are **…**:
- Not reported
- Yes, but they are not correct
- Yes, and they are correct

**Q4: **
Are the means and their MCSEs of alpha and beta in b) reported? Note that the number of digits reported for the means must follow the rule given in the assignment. The correct answers are alpha: mean **…** and beta: mean **…**.
- Not reported
- Yes, but one or both means are incorrect
- Yes, and the means are correct

**Q5: **
Are the quantiles and their MCSEs of alpha and beta in b) reported? Note that the number of digits reported for the quantiles must follow the rule given in the assignment. The correct answers are alpha: 5% quantile **…**, 95% quantile **…** and beta: 5% quantile **…**, 95% quantile **…**.
- Not reported
- Yes, but one or more quantiles are incorrect
- Yes, and the quantiles are correct

# Importance sampling

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

Implement a function for computing the log importance ratios (log importance weights) when the importance sampling **target distribution** is the posterior distribution, and the **proposal distribution** is the prior distribution from a). Explain in words why it’s better to compute log ratios instead of ratios.

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")`

.

Implement a function for computing normalized importance ratios from the unnormalized log ratios in c). In other words, exponentiate the log ratios and scale them such that they sum to one. Explain in words what is the effect of exponentiating and scaling so that sum is one.

Sample 4000 draws of \(\alpha\) and \(\beta\) from the prior distribution from a). Compute and plot a histogram of the 4000 normalized importance ratios. Use the functions you implemented in c) and d).

Use the function `rmvnorm`

from the `aaltobda`

package for sampling.

Using the importance ratios, compute the importance sampling effective sample size \(S_{\text{eff}}\) and report it.

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.*

Explain in your own words what the importance sampling effective sample size represents. Also explain how the effective sample size is seen in the histogram of the weights that you plotted in e).

Implement a function for computing the posterior mean using importance sampling, and compute the mean using your 4000 draws. Explain in your own words the computation for importance sampling. Report the means for \(\alpha\) and \(\beta\), and also the Monte Carlo standard errors (MCSEs) for the mean estimates. Report the number of digits for the means based on the MCSEs.

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 \(\text{Var}[\theta] = \text{E}[\theta^2] - \text{E}[\theta]^2\).

**Q6: **
Is the source code for the function in c) reported?
**Q7: **
Is the source code for the function in d) reported?
**Q8: **
Does the histogram in e) look something like this figure? If it is evident that the normalized importance ratios are computed correctly, but the prior was incorrect, you can still grade “Reported and looks similar”.
- Not reported
- Reported, but looks different
- Reported and looks similar

**Q9: **
Is the effective sample size in f) reported? The correct range for the effective sample size is between **…**. However, if it is evident that the effective sample size is computed correctly, but the prior was incorrect, you can still grade “Yes, and it is correct”.
- No
- Yes, but it is not correct
- Yes, and it is correct

**Q10: **
The correct explanation for g) is roughly the following: **…**
**Q11: **
What is the connection between S_eff and the histogram of weights: **…** How is the answer?
- Totally wrong/has not tried
- Something is a bit wrong
- Explanation is sensible

**Q12: **
Is the source code for the function in h) reported?
**Q13: **
Are the means and their MCSEs of alpha and beta in h) reported? Note that the number of digits reported for the means must follow the rule given in the assignment. The correct answers should be close to these: alpha: mean **…** and beta: mean **…**
- Not reported
- Yes, but they are incorrect
- Yes, and they are correct

# Overall quality of the report

**Q14: **
Does the report include comment on whether AI was used, and if AI was used, explanation on how it was used?
**Q15: **
Does the report follow the formatting instructions?
- Not at all
- Little
- Mostly
- Yes

**Q16: **
In case the report doesn’t fully follow the general and formatting instructions, specify the instructions that have not been followed. If applicable, specify the page of the report, where this difference is visible. This will help the other student to improve their reports so that they are easier to read and review. If applicable, specify the page of the report, where this difference in formatting is visible.
**Q17: **
Please also provide feedback on the presentation (e.g. text, layout, flow of the responses, figures, figure captions). Part of the course is practicing making data analysis reports. By providing feedback on the report presentation, other students can learn what they can improve or what they already did well. You should be able to provide constructive or positive feedback for all non-empty and non-nonsense reports. If you think the report is perfect, and you can’t come up with any suggestions how to improve, you can provide feedback on what you liked and why you think some part of the report is better than yours.