Monte Carlo Methods for Absolute Beginners

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Abstract

The modern history of Monte Carlo techniques dates back from the 1940’s and the Manhattan project. There are earlier descriptions of Monte Carlo experiments, Buffon’s famous needle experiment is one them, but examples have been traced back to Babylonian and old testament times [13]. As we shall see these techniques are particularly useful in scenarios where it is of interest to perform calculations that involve – explicitly or implicitly – a probability distribution π on a space X (typically X⊂Rnx for some integer nx), for which closed-form calculations cannot be carried out due to the algebraic complexity of the problem. As we shall see the main principle of Monte Carlo techniques consists of replacing the algebraic representation of π, e.g. 1/2π−−√exp(−12x2) with a sample or population representation of π, e.g. a set of samples X1,X2,…,XN∼iidπ(x)=1/2π−−√exp(−12x2). This proves in practice to be extremely powerful as difficult – if not impossible -exact algebraic calculations are typically replaced with simple calculations in the sample domain. One should however bear in mind that these are randomapproximations of the true quantity of interest. An important scenario where Monte Carlo methods can be of great help is when one is interested in evaluating expectations of functions, say f, of the type Eπ(f(X)) where π is the probability distributions that defines the expectation. The nature of the approach, where algebraic quantities are approximated by random quantities, requires one to quantify the random fluctuations around the true desired value. As we shall see, the power of Monte Carlo techniques lies in the fact that the rate at which the approximation converges towards the true value of interest is immune to the dimension nx of the space X where π is defined. This is the second interest of Monte Carlo techniques.
Translated title of the contributionMonte Carlo methods for absolute beginners
Original languageEnglish
Title of host publicationAdvanced Lectures on Machine Learning
Subtitle of host publicationML Summer Schools 2003, Canberra, Australia, February 2 - 14, 2003, Tübingen, Germany, August 4 - 16, 2003, Revised Lectures
PublisherSpringer Berlin Heidelberg
Pages113 - 145
Number of pages33
ISBN (Electronic)9783540286509
ISBN (Print)9783540231226
DOIs
Publication statusPublished - 2004

Publication series

NameLecture Notes in Computer Science
PublisherSpringer
Volume3176
ISSN (Print)0302-9743

Bibliographical note

Publisher: Springer-Verlag Berlin
Other identifier: IDS Number: BAX00

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