# Lectures on Monte Carlo Methods

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*Neal Madras*

A co-publication of the AMS and Fields Institute

Monte Carlo methods form an experimental branch of mathematics that employs
simulations driven by random number generators. These methods are often used
when others fail, since they are much less sensitive to the “curse of
dimensionality”, which plagues deterministic methods in problems with a
large number of variables. Monte Carlo methods are used in many fields:
mathematics, statistics, physics, chemistry, finance, computer science, and
biology, for instance.

This book is an introduction to Monte Carlo methods for anyone who would like
to use these methods to study various kinds of mathematical models that arise
in diverse areas of application. The book is based on lectures in a graduate
course given by the author. It examines theoretical properties of Monte Carlo
methods as well as practical issues concerning their computer implementation
and statistical analysis. The only formal prerequisite is an undergraduate
course in probability.

The book is intended to be accessible to students from a wide range of
scientific backgrounds. Rather than being a detailed treatise, it covers the
key topics of Monte Carlo methods to the depth necessary for a researcher to
design, implement, and analyze a full Monte Carlo study of a mathematical or
scientific problem. The ideas are illustrated with diverse running examples.
There are exercises sprinkled throughout the text. The topics covered include
computer generation of random variables, techniques and examples for variance
reduction of Monte Carlo estimates, Markov chain Monte Carlo, and statistical
analysis of Monte Carlo output.

Titles in this series are co-published with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

#### Reviews & Endorsements

Short but enlightening introduction to the subject … ideal text for a first graduate course on Monte Carlo methods for statisticians, mathematicians, computer scientists, and other scientists. The author is a leading expert in the field and he has made a careful choice of material to give readers a good basic introduction to the field.

-- CMS Notes

#### Table of Contents

# Table of Contents

## Lectures on Monte Carlo Methods

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Introduction 110
- Generating random numbers 1120
- Variance reduction techniques 2938
- Markov chain Monte Carlo 5362
- Statistical analysis of simulation output 8392
- The Ising model and related examples 93102
- Bibliography 101110
- Back Cover Back Cover1113