This document summarizes Monte Carlo integration techniques, including importance sampling and Markov chain Monte Carlo. It defines Monte Carlo integration as a numerical integration method using random numbers. The techniques were originally developed around 1944 to study the atomic bomb. Common uses are for evaluating integrals. The document outlines the basic steps of Monte Carlo methods and defines importance sampling as estimating properties of one distribution using samples from another similar distribution to reduce variance. It also defines Markov chain Monte Carlo as using a Markov chain to simulate samples from a target posterior probability distribution.

1. UNIVERSITY OF RWANDA
DEPARTMENT OF APPLIED STATISTICS
Monte Carlo integration, importance
sampling, basic idea of Markov
chain Monte Carlo.
BIZIMANA Appolinaire
Student in UR-College of Business
and Economics

2. This presentation is made up of
7 key points
I. Definition
II. History of Monte Carlo integration
III. The use of Monte Carlo integration
IV. How to apply of Monte Carlo integration
V. Importance sampling
VI. Basic idea of Markov chain Monte Carlo
(MCMC).
VII.Exercises of application to take a look at
them on word document.

3. I. Definition
It is somehow cumbersome to define Monte
carlo without taking into consideration on
its methods and domain of application, So
There is no consensus on how Monte
Carlo should be defined.
By definition In mathematics, Monte Carlo
integration is a technique for numerical
integration using random numbers. It
is a particular Monte Carlo method
that numerically computes a definite
integral.

4. II. History of Monte Carlo
integration
Monte Carlo methods were originally practiced
under more generic names such as
”statistical sampling”. The name ”Monte
Carlo” was popularized by physics researchers
S. Ulam, E. Fermi,J. von Neumann, and N.
Metropolis; The name is a reference to a
famous casino in Monaco where Ulam’s
uncle would borrow money to gamble. This
name and the systematic development of
Monte Carlo methods dates from about 1944.

5. III.The use of Monte Carlo
integration
The real use of Monte Carlo methods as a research tool
stems from work on the atomic bomb during the
second world war.
The most common use for Monte Carlo methods is
the evaluation of integrals.
IV. How to apply of Monte Carlo
integration
There are a number of practical methods that
use this principle to attempt to achieve better
estimates of the mean with fewer random
samples

6. Cont’d
We call these methods variance reduction techniques.
Those are practical techniques including adaptive
sampling, stratification, importance sampling, and
combined sampling.
Monte-Carlo methods generally follow the following steps
Determine the statistical properties of possible inputs
 Generate many sets of possible inputs which follows the
above properties
Perform a deterministic calculation with these sets
Analyze statistically the results.

7. V. Importance sampling
By definition is a general technique for estimating
properties of a particular distribution, while only
having samples generated from a different
distribution from the distribution of interest.
Importance sampling reduces variance by
observing that we have the freedom to choose
the PDF used during integration. By choosing
samples from a distribution pd f (x), which has a
similar shape as the function f (x) being
integrated, variance is reduced.

8. VII. Basic idea of Markov chain Monte
Carlo (MCMC).
In terms of meaning Markov chain Monte is a
mathematical model for stochastic system that
generates random variable X1, X2, X3,..........., Xt . P(Xt/
X1, X2, X3,.............,Xt-1). The distribution of the next
random variable depends on only the current
variable which the reason why of saying
chain.
So the Basic idea of Markov chain Monte Carlo
is the following, To construct a Markov Chain
Such that :

9. Cont’d
Have parameters as the state space, and
 the stationary distribution is posterior probability
distribution f the parameters.
Simulate the chain
Treat the realisation as a sample from the posterior
probability distribution
MCMC is a general purpose technique for generating
fair samples from a probability in high dimension
space, using random numbers draw from uniform
probability in certain range.

10. Conclusion
Monte Carlo simulations are an important tool in
modern-day studies of many physical systems.
Where unlikely events are to be simulated, the
importance sampling technique can considerably
ease the processing burden, without
compromising statistical significance. Here a
comparison of importance sampling and standard
Monte Carlo simulations is given. Emphasis is on
variance reduction, and on the simulation gain of
importance sampling.