lesson2-tasks.Rmd

---
title: "Lesson 2 - Tasks"
output: html_document
---


## Task 1: Prior predictive check

- Let us take a vague $N(0, 10)$ prior on the logarithm of the mean of a Poisson distribution. Run a prior predictive check, is there anything suspicious?
- Find a prior on the mean of Poisson distribution (or its logarithm - you pick) that matches this rough prior
knowledge: 
   - Zeroes are quite unlikely
   - Most values should be between 2 and 15
   - Values above 18 are quite unlikely 
 


## Task 2: Fitting a Poisson

- Build a model that takes an array of integers
- Accept arguments for prior for the logarithm of the mean as data
- Simulate 20 values from a Poisson distribution with known mean. Do you recover the values?

## Task 3: Detecting overdispersion with a posterior predictive check

- Simulate 30 values with neg. binomial with $\phi = \frac{1}{2}$ (`size` in R's `rnbinom`), fit with the model from Task 2. Do you recover the simulated mean of the neg. binomial?

- Try to detect the model-data mismatch with a posterior predictive check.
  

R:
  - Use `fit$draws(format = "draws_matrix")` to get a matrix of draws, then use `rpois`` to generate the predictions as the Stan model assumes
  - use either `bayesplot::ppc_stat` with a stat measuring the dispersion (e.g. variance, sd, ...) or `bayesplot::ppc_dens_overlay`
Python: use either  `arviz.plot_bpv()` with `kind="t_stat"`  and a stat measuring the dispersion (e.g. variance, sd, ...) or `arviz.plot_ppc`

- What is the smallest number of observations we need to reliably detect the problem?

- Implement a negative binomial model and see how the check behaves now. 
  - Use [`neg_binomial_2_lpmf`](https://mc-stan.org/docs/functions-reference/unbounded_discrete_distributions.html#nbalt))
  - Put an `exponential(1)` prior on the overdispersion parameter.

## Task 4: A bit more open-ended exploration

Given the following data:

```
y :  2, 0, 11, 25, 9, 4, 17, 11, 8, 4, 5, 2, 6, 4, 8, 24, 0, 3, 6, 4, 12, 9, 5, 6, 2, 10, 4, 15, 0, 1, 87, 19, 2, 1, 38, 16, 5, 7, 18, 11, 1, 0, 7, 15, 5, 0, 6, 1, 0, 6, 34, 29, 7, 11, 5, 10, 8, 3, 12, 18, 7, 1, 18, 18, 6, 3, 37, 5, 1, 0, 22, 13, 1, 0, 26, 19, 6, 7, 21, 45 
group :  "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B", "A", "B" 
type :  "X", "X", "Y", "Y", "X", "X", "Y", "Y", "X", "X", "Y", "Y", "X", "X", "Y", "Y", "X", "X", "Y", "Y", "X", "X", "Y", "Y", "X", "X", "Y", "Y", "X", "X", "Y", "Y", "X", "X", "Y", "Y", "X", "X", "Y", "Y", "X", "X", "Y", "Y", "X", "X", "Y", "Y", "X", "X", "Y", "Y", "X", "X", "Y", "Y", "X", "X", "Y", "Y", "X", "X", "Y", "Y", "X", "X", "Y", "Y", "X", "X", "Y", "Y", "X", "X", "Y", "Y", "X", "X", "Y", "Y" 
```

Try to fit the neg. binomial model from task 3 with a shared mean and overdispersion parameter to the `y` column of the data.

The model does not represent the data well. Try to figure out what it is and fix it!

Hint: many PPCs can be performed per group, in R this is the `ppc_xxx_grouped` functions.


## Task 5: Open-ended exploration 2

Given the following data:

```
y :  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1 
group :  "B", "C", "D", "B", "C", "B", "D", "C", "A", "D", "C", "C", "C", "D", "C", "C", "B", "B", "D", "C", "D", "C", "C", "A", "B", "D", "C", "D", "D", "D", "C", "D", "A", "A", "C", "D", "D", "C", "C", "B", "B", "C", "D", "D", "A", "D", "A", "B", "C", "A" 
```

Once again use the model from Task 3. Use posterior predictive check to determine what is wrong. 

Can you fix it?