Monte Carlo Methods

2 ECTS

SEMESTER 8

lectures

classes / seminars

practical work

integrative teaching

independent work

6 h

12 h

0 h

0 h

36 h

Language used

French

Course supervisor(s)

Key words

Random simulation; Monte Carlo method; Variance reduction; MCMC methods

Prerequisites

Probability theory (level: 1st year of master's degree); basics in Python programming language

Learn how to use random simulation to solve complicated deterministic problems.

   The first part of this course presents the Monte Carlo method to assess a probabilistic expectation by the use of simulations: by doing so, one computes a deterministic quantity through a random algorithm!

   In this first part, we will explain not only how to estimate the wanted quantity of interest; but also (and this is not less important) how to get the confidence interval for our estimator.  It may be important to be able to sharpen that confidence interval: that aim is achieved by the variance reduction techniques.  We will present four of these techniques, viz. importance sampling, conditioning, control variates and coupling (a.k.a. “common randomm numbers”).

   In many situations, simulating the needed wanted variables—which is of course central to the Monte Carlo method—turns out to be a more difficult challenge than applying the Monte Carlo method as such.  A typical situation where this occurs is when one needs to simulate random variables according to some posterior distribution in large-dimensional Bayesian statistics.  To tackle that issue, the second part of the course will present a range of methods to simulate variables following such random distributions, which methods are based on Markov chains, hence their name: Monte Carlo Markov Chain (or commonly “MCMC”) methods.

   In particular, we will study the Metropolis‒Hastings algorithm, which is very general; and we will also introduce the Gibbs sampling method, which is faster but also trickier to operate.  The reliability of such simulation methods (in particular concerning their speed of convergence) will be looked at under both theoretical and practical angles.

   A large part of the teaching will be devoted to implementing the studied concepts into computer programs.  For the computer exercise sessions, Python has been chosen for language.

Levels

Description and operational verbs

Know

How to compute the confidence interval for a Monte Carlo estimation method

What the main variance reduction techniques are

What are the formulas that characterize the Markov chains used in MCMC methods

Understand

What is the principle of the Monte Carlo method to estimate and expectation; What are its strengths and weaknesses

What is the use of the MCMC methods; How do the Markov chains involved in these methods behave

Apply 

Implement a Monte Carlo estimation method, resp. a variance reduction technique, into a computer program

Implement a Metropolis‒Hastings algorithm, resp. a Gibbs algorithm

Analyze 

Choose a reduction variance technique that will be appropriate for the problem considered

Look critically at the convergence behaviour of a Monte Carlo estimator

Choose an MCMC method that will be appropriate for the problem considered

Look critically at the convergence behaviour of a MCMC method

Summarize

In a statistical context, use an MCMC algorithm to compute relevant Bayesian estimators

Assess

Continuous assessment

Written test

Oral presentation / viva

Written report / project