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Bayesian Modelling B (MATH 34920)

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Administrative Information

  1. Unit number and title: MATH 34920 Bayesian Modelling B
  2. Level: H/6
  3. Credit point value: 10 credit points
  4. Year: 12/13
  5. First Given in this form: 2003-04
  6. Unit Organiser: David Leslie
  7. Lecturer: Dr D Leslie
  8. Teaching block: 2
  9. Prerequisites: MATH21400 Applied Probability 2 and MATH34910 Bayesian Modelling A

Unit aims

This unit will build on the material covered in Bayesian Modelling A, both by extending the range of models considered to include hierarchical specifications, and by deriving probabilistic algorithms that enable the practical use of Bayesian methods in a very broad range of applications.

General Description of the Unit

Much of the real advantage of the Bayesian approach to statistical modelling and inference, as compared to classical approach, is only seen when dealing with the slightly more complex situations encountered in this unit. Hierarchical models allow us to model situations where we simultaneously analyse different groups of data (for example, mortality statistics in different hospitals, or growth data in different children), and where the parameters describing the groups can be assumed to be similar - that is, not identical but not completely unrelated either.

To keep track of the different kinds of variation in these situations (for example, uncertainty in overall mortality, variation between hospitals, and variation among patients), it is useful to lay out the variables in a diagram, and the unit will include an introduction to these 'graphical models'.

We will go on to discuss how to draw inference in such models (answer - by Bayes' theorem!), and then how to actually do that in practice, since we will no longer have conjugacy to help us, as in Bayesian Modelling A. This leads to discussion of Markov chain Monte Carlo (MCMC) techniques, which are powerful and elegant algorithms based on simple ideas of conditional probability. Graphical modelling and MCMC are the basis for a package called WinBugs for doing Bayesian analysis without needing to write your own program, and there will be demonstrations and some hands-on practice with using that package on a range of interesting examples.

Relation to Other Units

The Level 7 units that build on the methods and knowledge discovered in Bayesian Modelling B are Monte Carlo Methods (M6001) and Graphical Modelling (M6002).

Teaching Methods

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

Learning Objectives

The students will be able to:

  • Represent complex data by means of a hierarchical model
  • Display such a model graphically
  • Understand and apply MCMC techniques for performing Bayesian analysis in practice
  • Justify theoretically the use of the various algorithms encountered

Assessment Methods

The assessment mark for Bayesian Modelling B is calculated from a 1½-hour written examination in May/June consisting of THREE questions. A candidate's best TWO answers will be used for assessment. Calculators are NOT permitted to be used in this examination.

Award of Credit Points

Credit points are gained by:

  • either passing the unit,
  • or getting a mark of 30 or over in the examination and handing in satisfactory attempts to all the set exercises.

Transferable Skills

In addition to the general skills associated with other mathematical units, you will also have the opportunity to gain practice in the following: computer literacy and general IT skills, use of R and WinBugs as programmable statistical packages, interpretation of computational results, time-management, independent thought and learning, and written communication.

Texts

The following texts may be useful for reference:

  1. Bernardo, J.M. and Smith, A.F.M. Bayesian Theory, John Wiley and Sons.
  2. Gamerman, D. Markov Chain Monte Carlo, Chapman and Hall.
  3. Gelman, A., Carlin, J.B., Stern, H.S. and Rubin, D.B. Bayesian Data Analysis, Chapman and Hall.
  4. Gilks, W.R., Richardson, S. and Spiegelhalter, D. Markov Chain Monte Carlo in Practice, Chapman and Hall.
  5. Morgan, B.J.T. Elements of Simulation, Chapman and Hall.
  6. Robert, C.P. The Bayesian Choice, Springer-Verlag.
  7. Robert, C.P. and Casella, G., Monte Carlo Statistical Methods, Springer-Verlag.

Syllabus

Hierarchical models; Directed acyclic graphs; Markov chain Monte Carlo; Gibbs sampler; Metropolis-Hastings algorithm; Application to analysing data, and posterior summaries.