# Monte Carlo Methods

## 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.

## 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.

## Syllabus

Introduction: motivating examples, random numbers, Monte Carlo integration. Fundamental concepts of transformation, rejection, and reweighting.

Elements of stochastic process theory for Markov chain Monte Carlo. Markov chains, stationary distributions, the intuition underlying convergence conditions.

Gibbs sampling and Metropolis-Hastings algorithms. Convergence diagnostics.

Structure of hidden Markov models, state-space models and the optimal filtering recursion.

Sequential Monte Carlo methods: sequential importance sampling, resampling. particle filtering.

Case studies and examples.

## Reading and References

- Gilks, W.R., Richardson, S. and Spiegelhalter, D. Markov Chain Monte Carlo in Practice, Chapman and Hall.
- Robert, C.P. and Casella, G., Monte Carlo Statistical Methods, Springer-Verlag.
- Jean-Michel Marin and Christian P. Robert, Bayesian Core: A Practical Approach to Computational Bayesian Statistics, Springer, to appear.
- Arnaud Doucet, Nando De Freitas and Neil J. Gordon (eds), Sequential Monte Carlo in Practice, Springer.
- Liu, J.S., Monte Carlo Strategies in Scientific Computing, Springer.

**Unit code:** MATHM6001

**Level of study:** M/7

**Credit points:** 10

**Teaching block (weeks):** 1 (7-12)

**Lecturer:** Dr Nick Whiteley

## Pre-requisites

MATH11300 Probability 1, MATH11400 Statistics 1 and MATH20008 Probability 2. MATH20800 Statistics 2 and MATH30015 Bayesian Modelling are desirable.

## Co-requisites

None

## Methods of teaching

Lectures, (theory and practical problems) supported by example sheets, some of which involve computer practical work with R or Matlab.

## Methods of Assessment

The pass mark for this unit is 50.

The final mark is calculated as follows:

- 80% from a 1 hour 30 minute exam in January
- 20% from homework assignments

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.