| Unit name | Monte Carlo Methods |
|---|---|
| Unit code | MATHM6001 |
| Credit points | 10 |
| Level of study | M/7 |
| Teaching block(s) |
Teaching Block 1A (weeks 1 - 6) |
| Unit director | Dr. Sejdinovic |
| Open unit status | Not open |
| Pre-requisites |
None |
| Co-requisites |
None |
| School/department | School of Mathematics |
| Faculty | Faculty of Science |
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, as the expected behaviour of a quantity related to some random phenomenon. 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.
Aims
The unit aims at providing students with sufficient background to undertake research in scientific areas that require the use of Monte Carlo methods. Although the emphasis will be on the application of the methods to Bayesian statistics, whenever possible other applications will be mentioned.
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.
Relation to Other Units
The unit aims to be self-contained: it does not require knowledge of any particular course in previous years, nor is it a pre-requisite for other courses. However, the introductions to Bayesian statistics given in "Statistics 2" and to the theory of Markov chains given in "Applied Probability 2" are relevant. Students may find that the Level 3 units "Bayesian Modelling A" and "Bayesian Modelling B", the Level M unit "Graphical Modelling", and the APTS courses on "Statistical Computing" and "Computer Intensive Statistics" complement this unit well.
The students will be able to:
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.
Lectures, (theory and practical problems) supported by example sheets, some of which involve computer practical work with R or Matlab.
The final assessment mark for Monte Carlo Methods is calculated as follows:
