# Assignment 2

# 1 General information

The exercises here refer to the lecture 2/BDA chapters 1-2 content. All questions check your understanding of a simple posterior analysis using a binomial model for the observations and a beta prior.

**The exercises constitute 90% of the Quiz 2 grade.**

We prepared a quarto notebook specific to this assignment to help you get started. __You still need to fill in your answers on Mycourses!__ You can inspect this and future templates

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## 1.1 Assignment questions

For convenience the assignment questions are copied below. Answer the questions in MyCourses.

**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 \( \theta \) be the probability of a monitoring site having detectable blue-green algae levels, \( y \) the number of observed sites with algae detected, and \( n \) be total number sites surveyed. Use a binomial model for the observations and a \( \text{Beta}(2,10) \) prior for binomial model parameter \( \theta \) to formulate a Bayesian model. Here we will not test you on the individual mathematical operations needed in order to derive the posterior distribution for \( \theta \) as it has already been done in the book (and lecture) so make sure to look that up.

Your task is to perform Bayesian inference for a binomial model and fill in the quiz below based on it.

For questions with checkboxes, more than one answer may be
correct.

##### 1. Formulating Probabilities

The `algae`

dataset contains the results of 274 measurements from Finnish lakes, with the following results:

- No Algae: 230 sites
- Algae: 44 sites

Our goal for the following set of questions is to find the formulation of the posterior using a binomial likeliood and a beta prior on the unknown probability parameter \( \theta \)

- 1.1 The prior \(p(\theta)\) can be expressed as:

- 1.2 The likelihood \( p(y = 44 | \theta, n = 274) \) as a function of \( \theta \) can be expressed as:

- 1.3 The resulting posterior \( p(\theta|y = 44, n = 274) \) can be expressed as :

##### 2. Summary of the posterior distribution of \( \theta \)

The posterior distribution \( p(\theta|y) \) is analytically available as \( \text{Beta}(\alpha, \beta) \), so we can use the properties of that distribution to summarise what we know about \( \theta \). And in particular, we can make probability statements about ranges of values for \( \theta \). Let's however start with the average value of \( \theta \) you expect after having conditioned on the data.

- 2.1 Which of the following is the correct formula for the mean (\( E[\cdot] \)) of a \( \text{Beta}(\alpha, \beta) \) distribution:

- 2.2 Using your answer above, what is the mean of our posterior (i.e., \( E(\theta|y) \))? Report the result in decimals with two decimal digits.

Posterior intervals are sometimes called credible intervals and are different from confidence intervals (for more on this, see here). These are computed using the quantile function of the posterior distribution. As the quantiles of a \( \text{Beta}(\alpha, \beta) \) distribution do not have a simple analytical form like the expectation, you can use R to compute the posterior intervals.

- 2.3 What R function would you use here to compute posterior intervals?

Using your answer above, calculate (report the results in decimals with two decimal digits):

- 2.4 90% posterior interval lower bound:

- 2.5 90% posterior interval upper bound:

##### 3. Comparison to historical records

We are interested in using our posterior distribution to estimate the probability that the proportion of detected algae samples (\( \theta \)) is smaller than the historical detection rate \( \theta_0 = 0.2 \), i.e. \( p(\theta \leq \theta_0 \mid y) \).

- 3.1 Which of the following approaches would we take?

- 3.2 What statistical function computes this probability for us?

- 3.3 Which R function does this for you?

- 3.4 Using your answers above, report this probability (report the result in decimals with two decimal digits):

##### 4. Prior sensitivity analysis

Redo the analysis using a uniform prior, \( \text{Beta}( 1,1 \)).

- 4.1 What is the mean of our posterior (i.e., \( E(\theta|y) \))? Report the result in decimals with two decimal digits.
- 4.2 90% posterior interval lower bound. Report the result in decimals with two decimal digits.
- 4.3 90% posterior interval upper bound. Report the result in decimals with two decimal digits.
- 4.4 Probability \( p(\theta \leq \theta_0 \mid y) \). Report the result in decimals with two decimal digits.

Redo the analysis using as prior \( \text{Beta}(0.5,0.5) \).

- 4.5 What is the mean of our posterior (i.e., \( E(\theta|y) \))? Report the result in decimals with two decimal digits.
- 4.6 90% posterior interval lower bound. Report the result in decimals with two decimal digits.
- 4.7 90% posterior interval upper bound. Report the result in decimals with two decimal digits.
- 4.8 Probability \( p(\theta \leq \theta_0 \mid y) \). Report the result in decimals with two decimal digits.

Redo the analysis using as prior \( \text{Beta}(100,2) \).

- 4.9 What is the mean of our posterior (i.e., \( E(\theta|y) \))? Report the result in decimals with two decimal digits.
- 4.10 90% posterior interval lower bound. Report the result in decimals with two decimal digits.
- 4.11 90% posterior interval upper bound. Report the result in decimals with two decimal digits.
- 4.12 Probability \( p(\theta \leq \theta_0 \mid y) \). Report the result in decimals with two decimal digits.