Exercises for Tue Mar 6

(Ved en s.k. inkurie er denne meldingen forsinket, og jeg beklager.)

1. Finalise the Bayesian nonparametric analysis of the Old Egypt life-times, men and women. Invent an interesting parameter or two, say \gamma(A_men, A_women), which in a suitable fashion helps to see the different by the two populations, and, with your own priors for A_men and A_women, produce the posterior distribution for \gamma.

2. Let A be a Beta process with parameters (c, A_0). On top of A, there is a Bernoulli process Z with A as parameter. This is taken to mean that Z given A has independent increments, with lots of 0 and occasional 1, and with dZ(s), again given A, a Bernoulli variable with probability dA(s). Show that A given Z is another Beta process, and identify its updated parameters.

3. Create an illustration of Exercise 2, in the following fashion. (i) Let A be a Beta process on say [0, 10], with A_0 the integral of \alpha_0(s) = 1 + a s, with some constant a to reflect higher expected intensity as time goes by. Also, let c(s) = c \exp(-d s), to reflect more prior uncertainty as time goes by. Simulate 25 realisations of such an A process. (ii) Let Z be a Bernoulli process, as described above, but with an underlying true A_\true intensity, with A_\true the integral of \alpha_\true(s) = 2 + 0.5 s. Simulate 25 realisations of such a Z process. (iii) With one of these, call them "real data", do the updating for A, and produce 25 realisations (or more) from the posterior distribution. 

I'm working on my Bragel?fte Nils Lecture Notes and Exercises, and should have some decent pages pretty soon, and with more to come.