Exercises for Tue Jan 23
1. We started the course Tue Jan 16, where I gave a broad introduction, partly via the "A gentle introduction to Bayesian Nonparametrics" pdf, and which I'll continue using next week. I defined the first Real Thing of the course, the Dirichlet process, with P \sim Dir(a P_0), which when relevant or practical is written F \sim Dir(a F_0). I dared to mention my Fame Parameter, which had a Pinker caused peak Mon Jan 15, and which via its World of Wars theme will also lead to a couple of exercises in this course, reasonably soon.
https://www.mn.uio.no/math/english/research/projects/focustat/the-focustat-blog!/krigogfred.html
2. On the course website there are now two documents which you should read through: the "Gentle introduction" (a talk I gave in Feb 2017) and the intro chapter to the Hjort, Holmes, Müller, Walker book (2008).
3. Pretty soon we start on Ch 2 of the course book Müller, Quintana, Jara, Hanson (2015), but we'll need some more time on "general introduction" matters.
4. Exercises for Tue Jan 23: (i) Go to the stk 4021 website 2017 to find the Nils Exercises collection there. Go through Exercise 13 there. (ii) Let P \sim Dir(aP_0), on some proper space. Show that V = P(A), with A a given subset, is a Beta (ap_0, a(1-p_0)), with p_0 = P_0(A). Find the mean and variance of V. (iii) With A and B two disjoint sets, find the covariance and correlation between P(A) and P(B). (iv) Now consider a Dir(a F_0) on the unit interval, with F_0(x) = x^2 and a = 3.333. Show that F_0 is a Beta(2,1) (so in your R program you can easily alter this to any Beta prior guess distribution). Set up a simulation scheme for drawing realisations from the random F, and produce a plot of ten such random cdfs. (v) Suppose you observe 0.103, 0.110, 0.140, 0.175, 0.186, 0.205, 0.219, 0.348, 0.511, 0.592 (which I have generated from another distribution, namely the Beta(1,2)). Set up a new simulation scheme for drawing realisations for the random F given these ten data points. Plot ten such curves. Then draw sim = 10^4 such curves in your computer, and display a histogram for the parameter \gamma = P(X in [0.2,0.3]) / P(X in [0.7,0.8]). (vi) Play with the parameters of the game to learn more about the Dirichlet process in practice.