Lesson 2 - Tasks

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 eitherbayesplot::ppc_statwith a stat measuring the dispersion (e.g. variance, sd, ...) orbayesplot::ppc_dens_overlayPython: use eitherarviz.plot_bpv()withkind=“t_stat”and a stat measuring the dispersion (e.g. variance, sd, ...) orarviz.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)
  • 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?