More HW hints, Ch4 Exercise 16
Many students have asked about part b).
The idea is this: how well are each of these three models able to accommodate the outliers present in the data? One way to assess this is by considering the posterior distribution of the width of an interval in the *likelihood.* Some models will shift the mean up to reach the outlier, while others will have sufficiently heavy tails to accommodate it in the variance.
So for each model's population-level sigma parameter (i.e., 1/sqrt(tau0[k] for k=1,2,3), we consider the posterior distribution of the width of a 95% interval in the likelihood that model induces for the data.
To get this posterior, we use the CODA samples of sigma[k]. For each Gibbs draw, compute the .025 and .975 quantiles of the associated likelihood (normal, t, or DE), take their difference, and call this a single posterior sample from the likelihood interval width. Repeat this procedure for all the sigma[k] Gibbs draws and you will have a histogram for the posterior of this quantile difference in the k^th model. Then you can compare across models.