--- title: 'Module 5: Exercises' author: "" date: "" output: html_document --- ## Data First recreate the same artificial dataset `x` as in the supplementary material (note the `set.seed()` command): ```{r} mu_true <- 250 tau_true <- 1/5^2 n <- 30 set.seed(42) x <- rnorm(n, mean = mu_true, sd = sqrt(1/tau_true)) x_bar <- mean(x) ``` ## Exercise 1 Now, assume the researcher a priori (before seeing any data/people) is sure that she/he is in a land of tiny people, and chooses the following parameters for the priors for the mean and precision (still product of normal and gamma distribution): ```{r} tau_prior <- 1/10^2 mu_prior <- 50 alpha_prior <- 10 beta_prior <- .002 ``` Rerun the Gibbs sampler from the supplementary material (with the same updating scheme: first $\mu$ then $\tau$) with these prior parameters and comment on the marginal distribution of $\mu$ (and $\tau$ if you like). ## Exercise 2 Now rerun the Gibbs sampler from the supplementary material with the opposite updating scheme -- first $\tau$ then $\mu$ -- and comment on the marginal distribution of $\mu$ (and $\tau$ if you like). ## Exercise 3 Try to explain the different results you obtained in exercises 1 and 2 by looking at the posterior we are trying to sample from. **Don't spend too much time on this** -- it is difficult, and you will probably have to consider the logarithm of the posterior to be able to identify the problem: Instead consult the solution if you find it too difficult. ## Exercise 4 Now generate additional 70 data points (people) from the population (still N(`r mu_true`, `r tau_true`)) and rerun the analysis from exercise 1 with this new dataset of 100 people (**remember** to update the global variables `n` and `x_bar`). Comment on the results. ## Exercise 5 Based on the posterior simulations in exercise 4 estimate the probability of $\mu < 249$.