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Machine Learning, 50, 5–43, 2003
c
°
2003 Kluwer Academic Publishers. Manufactured in The Netherlands.
An Introduction to MCMC for Machine Learning
CHRISTOPHE ANDRIEU
C.Andrieu@bristol.ac.uk
Department of Mathematics, Statistics Group, University of Bristol, University Walk, Bristol BS8 1TW, UK
NANDO DE FREITAS
nando@cs.ubc.ca
Department of Computer Science, University of British Columbia, 2366 Main Mall, Vancouver,
BC V6T 1Z4, Canada
ARNAUD DOUCET
doucet@ee.mu.oz.au
Department of Electrical and Electronic Engineering, University of Melbourne, Parkville, Victoria 3052, Australia
MICHAEL I. JORDAN
jordan@cs.berkeley.edu
Departments of Computer Science and Statistics, University of California at Berkeley, 387 Soda Hall, Berkeley,
CA 947201776, USA
Abstract.
This purpose of this introductory paper is threefold. First, it introduces the Monte Carlo method with
emphasis on probabilistic machine learning. Second, it reviews the main building blocks of modern Markov chain
Monte Carlo simulation, thereby providing and introduction to the remaining papers of this special issue. Lastly,
it discusses new interesting research horizons.
Keywords:
Markov chain Monte Carlo, MCMC, sampling, stochastic algorithms
1.
Introduction
A recent survey places the Metropolis algorithm among the ten algorithms that have had the
greatest in±uence on the development and practice of science and engineering in the 20th
century (Beichl & Sullivan, 2000). This algorithm is an instance of a large class of sampling
algorithms, known as Markov chain Monte Carlo (MCMC). These algorithms have played
a signi²cant role in statistics, econometrics, physics and computing science over the last
two decades. There are several highdimensional problems, such as computing the volume
of a convex body in
d
dimensions, for which MCMC simulation is the only known general
approach for providing a solution within a reasonable time (polynomial in
d
) (Dyer, Frieze,
& Kannan, 1991; Jerrum & Sinclair, 1996).
While convalescing from an illness in 1946, Stan Ulam was playing solitaire. It, then,
occurred to him to try to compute the chances that a particular solitaire laid out with 52 cards
would come out successfully (Eckhard, 1987). After attempting exhaustive combinatorial
calculations, he decided to go for the more practical approach of laying out several solitaires
at random and then observing and counting the number of successful plays. This idea of
selecting a statistical sample to approximate a hard combinatorial problem by a much
simpler problem is at the heart of modern Monte Carlo simulation.
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C. ANDRIEU ET AL.
Stan Ulam soon realised that computers could be used in this fashion to answer ques
tions of neutron diffusion and mathematical physics. He contacted John Von Neumann,
who understood the great potential of this idea. Over the next few years, Ulam and Von
Neumann developed many Monte Carlo algorithms, including importance sampling and
rejection sampling. Enrico Fermi in the 1930’s also used Monte Carlo in the calculation of
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 Spring '08
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 Machine Learning

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