Exercises for Tue Jan 30

1. On Jan 23 I went through more of the "gentle introduction to Bayesian Nonparametrics" material, and did also discuss how to simulate from the Dirichlet process, which is one of the basic tools of the course. See my "com1a" R script. Next week there'll be more about the Dirichlet process and its properties, before and after data.

2. Exercises for Jan 30 are as follows.

(i) Go to the krigogfred dataset (on the webpage), comprising (x, z) for 95 horrible wars, from 1823 to the 2003, with x = time of onset and z = the number of battle deaths. This is the set of all wars with z \ge 1000. Now form the subset of 51 wars where z \ge 7061, where the power law tail behaviour is meant to hold. This means that v_i = \log z_i - \log 7061 are seen as Expo(\theta). Do a simple ML analysis to see the size of \theta, and also a logLprofile to detect the Cunen-Hjort Vietnam Hypothesis of Peace-and-War statistics (read the FocuStat blog, which Pinker liked so much, etc.). Divided the v_i into 37 to the left, coming from F, and 14 to the right, coming from G. Carry out Bayesian nonparametrics!, with F from Dir(a F_0) and G from Dir(b G_0). Elicit good (a, F_0) and (b, G_0) for the purpose. Draw 25 F curves and 25 G curves, placed to the left and to the right on your screen and on your printout. Carry out inference for the median difference \gamma = F^{-1}(0.50) - G^{-1}(0.50), and translate it all to the z scale of battle deaths. Include also a few other relevant parameters to analyse and interpret. Contact NRK or CNN to convey your findings.

(ii) Let P be a Dir(a P_0), with P_0 being some continuous distribution, like the standard normal, and let X be drawn from P. Find the distribution for X. Next, let (X1, X2) be a sample of the enormous size 2 from P. Find the distribution of (X1, X2). Find in particular the probability that X2 = X1.

(iii) Let P be a Dir(a P_0) and X be a draw from P. Attempt to prove the Mother Theorem about the Dirichlet, that P given x is a Dir(a P_0 + \delta(x)).