Monte Carlo theory, methods and examples

I have a book in progress on Monte Carlo, quasi-Monte Carlo and Markov chain Monte Carlo. Several of the chapters are polished enough to place here. I'm interested in comments especially about errors or suggestions for references to include. There's no need to point out busted links (?? in LaTeX) because the computer will catch those for me when it is time to root out the last of them.

@book{mcbook,
   author = {Art B. Owen},
   year = 2013,
   title = {Monte Carlo theory, methods and examples}
}

Copyright Art Owen, 2009-2013,2018-2019.

Contents
  1. Introduction
  2. Simple Monte Carlo
  3. Uniform random numbers
  4. Non-uniform random numbers
  5. Random vectors and objects
  6. Processes
  7. Other integration methods
  8. Variance reduction
  9. Importance sampling
  10. Advanced variance reduction
  11. Markov chain Monte Carlo
  12. Gibbs sampler
  13. Adaptive and accelerated MCMC
  14. Sequential Monte Carlo
  15. Quasi-Monte Carlo
  16. Lattice rules
  17. Randomized quasi-Monte Carlo

Chapters 1 and 2

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


3 Uniform Random Numbers

  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

4 Non-uniform Random Numbers

  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

5 Random vectors and objects

  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
  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

6 Processes

  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

7 Other quadrature methods

  1. The midpoint rule
  2. Simpson's rule
  3. Higher order rules
  4. Fubini, Bahkvalov 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

8 Variance reduction

  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

9 Importance sampling

  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

10 Advanced variance reduction

  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 samplinlg
  8. Control variantes wtih antithetics and stratification
  9. Bridge, umbrella and path sampling
  10. End notes
  11. Exercises

15,16,17 QMC and RQMC chapters

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

16 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

Appendix A: The ANOVA decomposition of \([0,1]^d\)

  1. ANOVA for tabular data
  2. The functional ANOVA
  3. Orthogonalithy 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