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# Inference for binomial proportion

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 dataset `algae`

in the `aaltobda`

package (‘0’: no algae, ‘1’: algae present).

Let \(\pi\) 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 \(\BetaDist(2,10)\) prior for binomial model parameter \(\pi\) to formulate a Bayesian model. Here it is not necessary to derive the posterior distribution for \(\pi\) as it has already been done in the book and it suffices to refer to that derivation. Also, it is not necessary to write out the distributions; it is sufficient to use label-parameter format, e.g. \(\BetaDist(\alpha,\beta)\).

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

Formulate

- the likelihood \(p(y|\pi)\) as a function of \(\pi\),
- the prior \(p(\pi)\), and
- the resulting posterior \(p(\pi|y)\).

Report the posterior in the format \(\BetaDist(\alpha,\beta)\), where you replace \(\alpha\) and \(\beta\) with the correct numerical values.

With a conjugate prior, a closed-form posterior has Beta form (see equations in BDA3 and in the slides).

What can you say about the value of the unknown \(\pi\) according to the observations and your prior knowledge? Summarize your results with a point estimate (i.e. \(E(\pi|y)\)) and a 90% posterior interval.

Posterior intervals are also called credible intervals and are different from confidence intervals.

What is the probability that the proportion of monitoring sites with detectable algae levels \(\pi\) is smaller than \(\pi_0=0.2\) that is known from historical records?

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 BDA3 Chapter 5 and Lecture 7.

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.

**Q3: **
Is source code included?
**Q4: **
Are the prior, likelihood and posterior forms in a) reported (derivation of posterior not necessary)?
**Q5: **
Is the reported resulting posterior correct **…**?
- It is not reported, that the posterior distribution is a
**…** distribution.
- It is reported, that the posterior distribution is
**…**, but the numerical values for the parameters are incorrect
- It is reported, that the posterior distribution is
**…**, and the numerical values for the parameters are correct.

**Q6: **
In part b), is there at least one point estimate reported. Sample based estimates are also OK. Points should be given if the method is right, even if the result is wrong due to a wrong posterior distribution being used. With the right posterior, mean, median, and mode are all approximately **…**.
**Q7: **
For the b) part, is the 90% posterior interval reported? Sample based estimate is also OK. Points should be given if the method is right, even if the result is wrong because the posterior was wrong in the first place. If the posterior was right, the 90% posterior interval is roughly **…**.
**Q8: **
For the c) part, is the posterior probability Pr(π<0.2|y) reported? Points should be given if the method is right, even if the result is wrong because the posterior was wrong. If the posterior was right, the result should be approximately **…**.
**Q9: **
For the d) part, does the report discuss **…**
- No
- No, but other reasonable assumptions are discussed
- Yes, but not quite right or some missing
- Yes

**Q10: **
For the e) part, is there some comparison and discussion of results obtained with alternative prior parameters?
- No
- Yes, but the results and conclusions are clearly wrong
- Yes

# Overall quality of the report

**Q11: **
Does the report include comment on whether AI was used, and if AI was used, explanation on how it was used?
**Q12: **
Does the report follow the formatting instructions?
- Not at all
- Little
- Mostly
- Yes

**Q13: **
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.
**Q14: **
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.