Stat 202C: Computational Statistics, part 3

Monte Carlo Methods in Statistics

Monte Carlo methods have developed hand in hand with computers and are now an essential tool for scientific computations. This course deals with their relevance for statistics and will focus on three major applications of Monte Carlo methods in Statistics: the use of Markov chain Monte Carlo (MCMC) methods to explore posterior distributions derived in Bayesian approach, the Bootstrap, and permutation testing.
Below we indicate a tentative list of topics. As the quarter progresses we will identify the material covered in each week of instruction and list reading requirements.

1. The first electronic computer and the birth of Monte Carlo method. Contributions by Ulam and von Neumann.
   Using the computer to generate random numbers. Pseudo random numbers. Rejection sampling.
   Liu Ch 1, Sec 2.1, 2.2. R.C. Ch 1, 2.
2. Numerical integration. The curse of dimensionality. Importance sampling The principles of weighting and accept/reject. An approximation of the variance of Importance Sampling algorithm.
   Liu: Ch 2 as an overview. Details in Sections 2.5.1-2.5.4 and 2.6.2-2.6.3. R.C. 3.1-3.3
3. Introduction to sequential importance sampling. Fisher's exact test and the problem of exploring the space of all contingency tables with given margins. Permutation tests. A sequential importance sampling algorithm for the problem. Notion of volume tests.
   Liu: 3.4; R.C. take a look at Ch. 14. Web references P. Diaconis and B. Efron (1985); Chen, Diaconis, Holmes and Liu (2005).
4. Markov chains; ergodic theorems for MC with finite state space.
   Liu: Ch 12: 1-3. R. C. Ch 6 (obviously there is much more here than what we covered in class)
5. Designing a Markov chain with a given invariant distribution: Metropolis algorithm and variations.
   Liu: Ch 5. R. C. Ch 7, Web references : Cipra's introduction to the Ising Model
6. Gibbs sampler. Liu: Ch 6.1-4, R. C. Ch 9.1-9.1.3; 10.1; 10.2.2
7. Rate of convergence of Markov Chains: few results. The interest of autocorrelation.
   Liu: Ch 6.6, Ch. 12.6.
8. Grouping, collapsing, auxiliary variables, and simulated tempering.
   Liu: Ch 6.7, Ch 7. Web references : Geyer (1991), Geyer and Thompson (1995), Swendsen and Wang (1987)
9. Bootstrap method for variance estimation and testing.
   Hand out from Efron and Tibshirani distributed in class.
10. Exact sampling, coupling from the past.
    R. C. Ch 13 . Web references : Propp and Wilson (1998).