Skip to main content

# Unit information: Monte Carlo Methods in 2020/21

Unit name Monte Carlo Methods MATHM6001 10 M/7 Teaching Block 2C (weeks 13 - 18) Professor. Andrieu Not open MATH11300 Probability 1, MATH11400 Statistics 1 and MATH20008 Probability 2 MATH20800 Statistics 2 and MATH30015 Bayesian Modelling are desirable but not essential None School of Mathematics Faculty of Science

## Description

Unit Aims

The unit aims to provide students with sufficient background to undertake research in scientific areas that require the use of Monte Carlo methods, by equipping them with the knoweldge and skills to understand, design and apply these techniques. Applications to Bayesian statistics will be discussed.

Unit Description

Modern statistics and connected areas very often require the numerical approximation of quantities that are crucial to the understanding of scientific problems as diverse as robot navigation target tracking, wireless communications, epidemiology or genomics to name a few. The Monte Carlo method can be traced back to Babylonian and Old Testament times, but has been systematically used and known under this name since the times of the "Los Alamos School" of physicists and mathematicians in the 1940's-50's. The method is by nature probabilistic and has proved to be a very efficient tool to approximate quantities of interest in various scientific areas.

The main idea of Monte Carlo methods consists of reinterpreting mathematical objects, e.g. an integral or a partial differential equation, in terms of the expected behaviour of a random quantity. For example p = 3.14 can be thought of as being four times the probability that raindrops falling uniformly on a 2cmx2cm square hit an inscribed disc of radius 1cm. Hence provided that realisations (drops in the example) of the random process (here the uniform rain) can be observed, it is then possible estimate the quantity of interest by simple averaging.

The unit will consist of: (i) showing how numerous important quantities of interest in mathematics and related areas can be related to random processes, and (ii) the description of general probabilistic methods that allow one to simulate realisations of such processes on a standard PC.

Relation to Other Units

Part of this unit expands upon and applies some of the Markov chain theory studied in Probability 2. The introductions to Bayesian statistics given in Statistics 2 and Bayesian Modelling will be very useful for students taking this unit.

## Intended learning outcomes

Learning Objectives

The students will be able to:

• Read and understand the scientific literature where standard Monte Carlo methods are used.
• Understand and develop Monte Carlo techniques for solving scientific problems, including Bayesian analysis.
• Understand the probabilistic underpinnings of the methods and be able to justify theoretically the use of the various algorithms encountered.

Transferable Skills:

In addition to the general skills associated with other mathematical units, students will also have the opportunity to gain practice in the implementation of algorithms in R.

## Teaching details

The unit will be taught through a combination of

• synchronous online and, if subsequently possible, face-to-face lectures
• asynchronous online materials, including narrated presentations and worked examples
• guided asynchronous independent activities such as problem sheets and/or other exercises
• synchronous weekly group problem/example classes, workshops and/or tutorials
• synchronous weekly group tutorials
• synchronous weekly office hours

## Assessment Details

80% Timed, open-book examination 20% Coursework

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

## Reading and References

Recommended

• Arnaud Doucet, Nando De Freitas and Neil Gordon (eds), Sequential Monte Carlo in Practice, Springer, 2001
• Walter R. Gilks, Sylvia Richardson and D.J. Spiegelhalter, Markov Chain Monte Carlo in Practice, Chapman and Hall, 1996
• Jun S. Liu, Monte Carlo Strategies in Scientific Computing, Springer, 2008
• Jean-Michel Marin and Christian P. Robert, Bayesian Core: A Practical Approach to Computational Bayesian Statistics, Springer, 2007
• Christian P. Robert and George Casella, Monte Carlo Statistical Methods, Springer-Verlag, 2004