Exercises for Tue Feb 27

1. On Tue Feb 20 we had a good and interesting round on Gamma processes, extended Gamma processes, Levy representations, and Nils' Beta processes. An idea was thrown out that we could make an extended Gamma process generating increments as parameters for a nonhomogeneous Poisson process, and which should be updatable. Emil S has pursued this idea. I also indicated how a prior Beta process (c, A_0) is updated to a posterior Beta process (c_n, \hat A_n), after observing a survival dataset (t_1,\delta_1), ..., (t_n,\delta_n).

2. I've updated (a) the R script com6a and (b) the dataset egypt_data to the course website. com6a does prior and posterior Beta process simulation; egypt_data has lifelengths for 82 men and 59 women, from the first century B.C.

3. Exercises:

(i) First, run the com6a programme, playing with its ingredients, including the prior parameters (c, A_0) of the Beta process.

(ii) Then, use variations o fthis programme to analyse the Egypt data, with men and women separately. Invent your own decent priors. Simulate cumulative hazard rates A(t) and survival curves S(t), from your priors, and from the posterior Beta processes, for men and women separately.  

(iii) Use Bayesian nonparametrics via these Beta processes to analyse S_men(t) - S_women(t), along with other parameters of relevance.

4. Nils bragepromises to write up several pages of Nils Exercises and Lecture Notes, pretty soon.