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Monte Carlo Methods (MATH M6001)
Contents of this document:
- Administrative information
- Unit aims, General Description, and Relation to Other Units
- Teaching methods and Learning objectives
- Assessment methods and Award of credit points
- Transferable skills
- Texts and Syllabus
Administrative Information
- Unit number and title: MATH M6001 Monte Carlo Methods
- Level: M/7
- Credit point value: 10 credit points
- Year: 12/13
- First Given in this form: October 2007
- Unit Organiser: Nick Whiteley
- Lecturer: Nick Whiteley
- Teaching block: 1
- Prerequisites: MATH11300 Probability 1 and MATH 11400 Statistics 1. MATH20800 Statistics 2 and MATH21400 Applied Probability 2 are desirable.
Unit 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.
General Description of the Unit
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
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 6 units "Bayesian Modelling A" and "Bayesian Modelling B", the Level 7 unit "Graphical Modelling", and the APTS courses on "Statistical Computing" and "Computer Intensive Statistics" complement this unit well.
Teaching Methods
Lectures, (theory and practical problems) supported by example sheets, some of which involve computer practical work with R or Matlab.
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.
Assessment Methods
The final assessment mark for Monte Carlo Methods is calculated as follows:
- 20% from homework.
- 80% from a standard, closed book 1 1/2 hour examination in April consisting of THREE questions. A candidate's best TWO answers will be used for assessment. Calculators are NOT allowed in this examination.
Award of Credit Points
Credit points will be awarded if a student passes the unit; i.e. attains a final assessment mark of 50 or more. If a student obtains a final assessment mark between 30 and 49, they may still be awarded the credit points if they have attended at least 90% of the lectures.
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.
Texts
- 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.
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.
Course Website
Further information about this course can be found at: