Monte Carlo theory, methods and examples

1 · Introduction

1. Example: traffic modeling
2. Example: interpoint distances
3. Notation
4. Outline of the book
5. End notes
6. Exercises

2 · Simple Monte Carlo

1. Accuracy of simple Monte Carlo
2. Error estimation
3. Safely computing the standard error
4. Estimating probabilities
5. Estimating quantiles
6. Random sample size
7. When Monte Carlo fails
8. Chebychev and Hoeffding intervals
9. End notes
10. Exercises

1. Random and pseudo-random numbers
2. States, periods, seeds, and streams
3. U(0,1) random variables
4. Inside a random number generator
5. Uniformity measures
6. Statistical tests of random numbers
7. Pairwise independent random numbers
8. End notes
9. Exercises

1. Inverting the CDF
2. Examples of inversion
3. Inversion for the normal distribution
4. Inversion for discrete random variables
5. Numerical inversion
6. Other transformations
7. Acceptance-rejection
8. Gamma random variables
9. Mixtures and automatic generators
10. End notes
11. Exercises

1. Generalizations of one-dimensional methods
2. Multivariate normal and t
3. Multinomial
4. Dirichlet
5. Multivariate Poisson and other distributions
6. Copula-marginal sampling  (home-made Gaussian copula)
7. Random points on the sphere
8. Random matrices
9. Example: classification error rates
10. Random permutations
11. Sampling without replacement
12. Random graphs
13. End notes
14. Exercises

1. Stochastic process definitions
2. Discrete time random walks
3. Gaussian processes
4. Detailed simulation of Brownian motion
5. Stochastic differential equations
6. Non-Poisson point processes
7. Dirichlet processes
8. Discrete state, continuous time processes
9. End notes
10. Exercises

1. The midpoint rule
2. Simpson's rule
3. Higher order rules
4. Fubini, Bakhvalov and the curse of dimensionality
5. Hybrids with Monte Carlo
6. Laplace approximations
7. Weighted spaces and tractability
8. Sparse grids
9. End notes
10. Exercises

1. Overview of variance reduction
2. Antithetics
3. Example: expected log return
4. Stratification
5. Example: stratified compound Poisson
6. Common random numbers
7. Conditioning
8. Example: maximum Dirichlet
9. Control variates
10. Moment matching and reweighting
11. End notes
12. Exercises

1. Basic importance sampling
2. Self-normalized importance sampling
3. Importance sampling diagnostics
4. Example: PERT
5. Importance sampling versus acceptance-rejection
6. Exponential tilting
7. Modes and Hessians
8. General variables and stochastic processes
9. Control variates in importance sampling
10. Mixture importance sampling
11. Multiple importance sampling
12. Positivisation
13. What-if simulations
14. End notes
15. Exercises

1. Grid-based stratification
2. Stratification and antithetics
3. Latin hypercube sampling
4. Orthogonal array sampling
5. Adaptive importance sampling
6. Nonparametric AIS
7. Generalized antithetic sampling
8. Control variates with antithetics and stratification
9. Bridge, umbrella and path sampling
10. End notes
11. Exercises

11 · Markov chain Monte Carlo

1. The need for MCMC
2. Markov chains
3. Detailed balance
4. Metropolis-Hastings
5. Random walk Metropolis
6. Independence sampler
7. Random disks revisited
8. Ising revisited
9. New proposals from old
10. Burn-in
11. Convergence diagnostics
12. Error estimation
13. Thinning
14. End notes
15. Exercises

12 · Gibbs Sampler

1. Stationary distribution for Gibbs
2. Example: truncated normal
3. Example: probit model
4. Aperiodicity, irreducibility, detailed balance
5. Correlated components
6. Gibbs for mixture models
7. Example: 10,000 galaxy velocities
8. Label switching
9. The slice sampler
10. Thinning
11. End notes
12. Exercises

13 · More MCMC methods — in progress.

14 · Some theory of MCMC — in progress.

15 · Quasi-Monte Carlo

1. Introduction to QMC
2. Discrepancy measures
3. Discrepancy rates
4. The Koksma-Hlawka Inequality
5. van der Corput and Halton sequences
6. Example: the wing weight function
7. Digital nets and sequences
8. Effect of projections
9. Example: synthetic integrands
10. How digital constructions work
11. Infinite variation
12. Higher order nets
13. Haar wavelets and Walsh functions
14. Kronecker sequences
15. End notes
16. Exercises

16 · Lattice rules

1. Grid-based stratification
2. Rank one lattices
3. Example: wing weight revisited
4. Lattices and lattice rules
5. Quality criteria for lattices
6. Convergence rates
7. Periodizing transformations
8. Lattice parameter search
9. Embedded, extensible and shifted lattices
10. Weighted spaces
11. End notes
12. Exercises

17 · Randomized quasi-Monte Carlo

1. Randomized quasi-Monte Carlo
2. RQMC definitions and basic properties
3. Effective dimension for RQMC
4. Cranley-Patterson rotation and lattices
5. Example: wing weight function
6. Scrambled nets
7. More scrambles
8. Reducing effective dimension
9. Example: valuing an Asian option
10. Padding, hybrids and supercube sampling
11. Randomized Halton sequences
12. RQMC and variance reduction
13. Singular integrands
14. (R)QMC for MCMC
15. Array-RQMC
16. End notes
17. Exercises

1. ANOVA for tabular data
2. The functional ANOVA
3. Orthogonality of ANOVA terms
4. Best approximation by ANOVA
5. Effective dimension
6. Sobol' indices and mean dimension
7. Anchored decompositions
8. End notes
9. Exercises