Course title:Monte Carlo Statistical Methods

Instructor:Sujit K. Ghosh, NC State University, USA.

Description:Monte Carlo (MC) statistical methods are used in statistics and various other fields (mathematics, physics, etc.) to solve various problems by generating suitable random numbers and observing that fraction of the numbers obeying some properties. The method is useful for obtaining numerical solutions to problems which are too complicated to solve analytically. The most common application of the Monte Carlo method is Monte Carlo integration. Deterministic methods of numerical integration operate by taking a number of evenly spaced samples from a function. In general, this works very well for functions of one variable. However, for functions of vectors, deterministic quadrature methods can be very inefficient. Monte Carlo methods provide a way out of this curse of dimensionality. As long as the function in question is reasonably well-behaved, it can be estimated by randomly selecting points in a high-dimensional space, and taking some kind of average of the function values at these points. Another powerful and very popular application for random numbers in numerical simulation is in numerical optimization. These problems use functions of some often large-dimensional vector that are to be minimized. Most Monte Carlo optimization methods are based on random walks. Essentially, the program will move around a marker in multi-dimensional space, tending to move in directions which lead to a lower function, but sometimes moving against the gradient.

Interestingly, the Monte Carlo method does not require truly random numbers to be useful. Much of the most useful techniques use deterministic, pseudo-random sequences, making it easy to test and re-run simulations. The only quality usually necessary to make good simulations is for the pseudo-random sequence to appear "random enough" in a certain sense.

Monte Carlo methods were originally practiced under more generic names such as "statistical sampling". The "Monte Carlo" designation, popularized by early pioneers in the field (including Stanislaw Marcin Ulam, Enrico Fermi, John von Neumann and Nicholas Metropolis), is a reference to the famous casino in Monaco. Its use of randomness and the repetitive nature of the process are analogous to the activities conducted at a casino. Stanislaw Marcin Ulam tells in his autobiography Adventures of a Mathematician that the method was named in honor of his uncle, who was a gambler, at the suggestion of Metropolis.

This course is intended to bring MC techniques into the classroom for a second-year graduate course as a self contained logical development of the subject, with all concepts being explained in detail. The course does not assume that the participants have any familiarity with MC methods, such as random number generation. The course does assume that the participant has familiarity with basic theoretical statistical concepts such as densities, distributions, probability and expectations, the Law of Large Numbers and the Central LimitTheorem. While this is a course on simulation, no requirement is made on programming skills or computing abilities; algorithms will be presented in class using the software R ( We strongly urge each participant to bring a (laptop) computer that has a copy of R installed on it. There will be a number of examples worked outin class.

A tentative outline of the syllabus of the course is given below:

  1. Introduction
  2. Scope of Course
  3. The Software R
  4. Computers as Inference Machines
  5. Deterministic Numerical Methods
  6. Optimization: Conjugate-gradient Methods
  7. Integration: Gaussian-quadrature Methods
  8. Examples using R (optim, integrate)
  1. Simulation
  2. Introduction
  3. Random Sampling using R
  4. Random Variables from Specified Distributions
  5. Inverse Transform
  6. Rejection Sampling (RS)
  7. Examples using R
  1. Monte Carlo Integration
  2. Introduction
  3. Strong Law of Large Numbers (SLLN)
  4. Central Limit Theorem (CLT)
  5. Importance Sampling (IS)
  6. Finite Variance Estimators
  7. Comparing RS and IS
  8. Examples using R
  1. Monte Carlo Optimization
  2. The EM Algorithm
  3. Monte Carlo EM
  4. EM Standard Errors
  5. Stochastic Exploration (optional)
  6. Simulated Annealing
  7. Prior Feedback Method
  8. Examples using R
  1. Markov Chain Monte Carlo (MCMC)
  2. The MCMC Principle
  3. The Metropolis-Hastings Algorithm
  4. The Gibbs Sampler
  5. The Slice Sampler
  6. Examples using R (MCMCpack) and WinBUGS (optional)

Course Schedule (Dec 17-29, 2007)

Dec 17
(Mon) / 9:00-12:00
1.00-04.00 /

Monte Carlo (MC) methods: An overview

Applications and case studies

Dec 18

(Tue) / 9:00-12:00
1.00-04.00 / Random variable generation

Illustrations using R

Dec 19

(Wed) / 9:00-12:00
1.00-04.00 / MC integration: Theory
Illustrations using R
Dec 20
(Thu) / 9:00-12:00
1.00-04.00 / MC optimization: Theory

Illustrations using R

Dec 21
(Fri) / 9:00-12:00
1.00-04.00 / Bootstrap methods: Theory
Illustrations using R
Dec 24
(Mon) / 9:00-12:00
1.00-04.00 / no class
no class

Dec 25

(Tue) / 9:00-12:00
1.00-04.00 / no class
Statistical simulation principle and Illustrations using R

Dec 26

(Wed) / 9:00-12:00
1.00-04.00 / Markov Chains: Theory
Markov Chain Monte Carlo (MCMC) principle
Dec 27
(Thu) / 9:00-12:00
1.00-04.00 /

Metropolis-Hastings algorithm: Theory

Illustrations using R
Dec 28
(Fri) / 9:00-12:00
1.00-04.00 /

Slice and Gibbs sampler: Theory

WinBUGS software

Dec 29

(Sat) / 9:00-12:00
1.00-04.00 / Illustrations using WinBUGS
no class

Comments:The first day (Dec 17th) presents an overview of the Monte Carlo methods and the lectures are suitable to general audience.

Dr. Sujit K Ghosh

Professor

Department of Statistics

North Carolina State University, USA.