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; Reduction of variance; MCMC methods

Prerequisites

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

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

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

   In this first part, not only one explains how to estimate the wanted quantity of interest; but also (and this is not less important) how to get the confidence interval for that estimator. It may be important to be able to sharpen that confidence interval: that aim is achieved by the techniques of reduction of variance. We will present four of these techniques: importance sampling, conditioning, control variables 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 issue than applying the Monte Carlo method as such. A typical situation where this occurs is when one wants to simulate a 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 chain, hence their name: Monte Carlo Markov Chain (commonly called “MCMC”) methods.

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

   A large part of this teaching will be devoted to implementing the studied concepts into computer programs: I chose the Python language for the computer exercise sessions.

Levels

Description and operational verbs

Know

Understand

Apply 

Analyse 

Summarise

Assess

Continuous assessment

Written test

Oral presentation / viva

Written report / project