exercises for Wed Nov 3

1. Due to hitches & glitches in the Bureacracy of Things, it's possible that there will be only Exam Part One, the 4-hour no-book exam on Nov 30, i.e. no Exam Part Two, the intended home project. Apologies for this not being crystal clear from day zero. A final message regarding this will be given next week.

2. Make The Oblig your priority, with submission deadline Tue Nov 2. I shall have time to skim through your reports so that I can say something about them on Wed Nov 3.

3. I hope you have time to work through the following "prototype exercise for Chapter 5 terrain", which we'll do before we take the Exam stk 4020 2012 no. 3, which is of the same type, but with more complexities involved.

These are your n = 40 data points, sorted here from small to big:

 0  2  2  3  3  3  3  3  3  3 

4  4  5  5  5  5  5  6  6  6 

6  6  6  7  7 7  7  7  8  8 

9  9 10 10 11 12 12 12 15 18

Take them as independent Poisson with parameters \theta_1, ..., \theta_{40}. Take these to have arisen from a gamma(a,b), with a flat prior for (a,b). Find ML estimators (\hat a,\hat b), using the marginal distribution f_\marg(\data, a,b). Then do MCMC for (a,b), and finally answer three natural questions.

(i) Give a figure with raw data y_i along with 0.05, 0.50, 0.95 posterior probability for the 40 parameters.

(ii) Give a figure of the underlying gamma density, say g(\theta, a,b), which we take to have generated the 40 \theta_i. Find a way to explain the uncertainty in this picture.

(iii) Give the predictive distribution for the next data point, say y_{41}.