PhD course 2006, Stochastic Simulation and Examples of Applications

1. Organiser and lecturer: Professor Jesper Møller, Department of Mathematical Sciences, Aalborg University (jm@math.aau.dk; www.math.aau.dk/~jm).
2. Place: Fredrik Bajers Vej 7G, Room G5-109, Aalborg University.
3. Course material: the following lecture notes and exercises which you are supposed to download and print before joining the course.
   1. Basics of probability theory.
   2. Exercises to Basics of probability theory.
   3. A very quick introduction to R.
   4. Basic methods for simulation of random variables: 1. Inversion.
   5. A brief introduction to (simulation based) Bayesian inference.
   6. Basic methods for simulation of random variables: 2. Accept/rejection algorithm.
   7. Simulation from specific distributions - especially the normal distribution.
   8. Monte Carlo methods.
   9. Model checking based on p-values.
   10. Some elementary Markov chain theory.
   11. A short diversion into the theory of Markov chains, with a view to Markov chain Monte Carlo methods.
   12. Importance sampling for unnormalized densities
4. Exercises and computer demos: The course material has previously been used for a PhD course of 5 ECTS, while the present course only counts as 4 ECTS. Consequently, I have decided in the present course not to talk about pseudo random number generators; in case you want to learn something about this topic, you may download an exercise on pseudo random number generators. Moreover, we will not have time to get you very familiar with the free software package R, which we will use for computer demos. In many cases when you are asked in an exercise to make some coding using R, you may instead download some code I have made (see "Programme" below) and discuss and use this code in order to illustrate the theory. Furthermore, you are very welcome on your own to solve these questions from the exercises, using either R or the software you are used to use. In fact you will learn a lot from doing so.
5. Some useful textbooks:
   Dani Gamerman, Markov Chain Monte Carlo, Chapman & Hall, London, 1997.
   Christian P. Robert and George Casella, Monte Carlo Statistical Methods, Springer-Verlag, New York, 1999.
   Jun S. Liu, Monte Carlo Strategies in Scientific Computing, Springer-Verlag, New York, 2001.
   Olle Häggström, Finite Markov Chains and Algorithmic Applications, Cambridge University Press, 2002.
   Andrew Gelman, John B. Carlin, Hal S. Stern and Donald B. Rubin, Bayesian Data Analysis, Second Edition, Chapman & Hall /CRC, Boca Raton, 2003.
   Geof H. Givens and Jennifer A. Hoeting, Computational Statistics, Wiley, Hoboken, New Jersey, 2005.
   

This is a highly intensive PhD course on which you should expect to spend all your working effort during the period 13-24 February. Moreover, you should prepare yourself before joining each day's programme, including the first time (Monday 13 February). A tentative programme is given below; it may very well be adjusted as the course goes on! The R codes mentioned are those we will use for the computer demos.

1) Monday 13 February 9:15-16:15

1. "Basics of probability theory."
2. "Exercises to Basics of probability theory."
3. First, I'll briefly tell you about "A very quick introduction to R." The message will be that we'll learn R by doing. Next, you are supposed to repeat doing the demo. In case you want (later and on your own!) to learn more about R, you may consult www.math.aau.dk/~slb/kurser/rbugs-05 (unfortunately, this site is in Danish), the first exercise on R and the second exercise on R.
4. "Basic methods for simulation of random variables: 1. Inversion." where we skip Exercise 4 (R codes to Exercises 3.5, 3.6, 3.7, 4.2, 5.1, 5.4, 5.5 and 5.6.)

1. First we consider again Exercise 5 from "Basic methods for simulation of random variables: 1. Inversion" and next we turn to "A brief introduction to (simulation based) Bayesian inference."
2. "Basic methods for simulation of random variables: 2. Accept/rejection algorithm." (R codes to Exercises 2.3, 3.1 and 3.5.)
3. "Simulation from specific distributions - especially the normal distribution." (R codes to Exercises 1.2 and 2.4.)
4. "Monte Carlo methods." In connection to Exercise 2 we visit www.mste.uiuc.edu/reese/buffon/buffon.html and www.angelfire.com/wa/hurben/buff.html. (R codes to Exercises 1.3, 1.5 and 3.5.)

1. "Model checking based on p-values." (R codes to Exercises 1, 2 and 3.)
2. "Some elementary Markov chain theory" where you should notice the following.
   1. In Exercise 2.5, please ignore the comment" by arguing as in the proof of Theorem 2.1, since we will skip (almost) all proofs in the text.
   2. In chapter 4, we skip Lemma 4.1, Corollary 4.1 and Exercise 4.3.
   3. Chapter 5 looks rather long, but since we skip the proofs, we are quickly through this. We also drop Exercises 5.1 and 5.5.
3. Earlier I wrote "Question: notice the programme for 24 February, where it is suggested that we may spend some time on discussing the participants own research problems in connection to stochastic simulation. Is that a good idea? Do we expect to have time enough?" I think the answer now is that we don't.

1. We review "Some elementary Markov chain theory" and solve the remaining exercises in Chapter 5.
2. Sections 1-4 in "A short diversion into the theory of Markov chains, with a view to Markov chain Monte Carlo methods." (R codes to Exercises 1.2, 1.3, 3.2, 5.2 and 6.2.)

1. We review Sections 1-4 in "A short diversion into the theory of Markov chains, with a view to Markov chain Monte Carlo methods."
2. Sections 5-9 in "A short diversion into the theory of Markov chains, with a view to Markov chain Monte Carlo methods." (R codes to Exercises 8.1, 8.2, 9.1, 9.2 and 10.)
3. In connection to Section 5, we also consider "Importance sampling for unnormalized densities."

1. We start by discussing Exercise 10 from Wednesday.
2. Section 10 and further on in "A short diversion into the theory of Markov chains, with a view to Markov chain Monte Carlo methods." At the end of Section 11.1 you are supposed to make some R-code based on modifying this R-code and concerning a simple image restoration problem (more details at the lecture.....). We will stop after Section 11.1.
3. Finally, we should also leave some time for discussing/evaluating this course.