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Graphical Modelling (MATH M6002)

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Administrative Information

  1. Unit number and title: MATH M6002 Graphical Modelling
  2. Level: M/7
  3. Credit point value: 10 credit points
  4. Year: 12/13
  5. First Given in this form: 2007-08
  6. Unit Organiser: Vanessa Didelez
  7. Lecturer: Dr V Didelez
  8. Teaching block: 1
  9. Prerequisites: MATH20800 Statistics 2 and MATH34910 Bayesian Modelling A

Unit aims

This unit is held in weeks 7-12 of Teaching Block 1

This unit will introduce the theory of graphical modelling for complex statistical models, and will cover the practical use of this theory in several areas including, but not necessarily limited to: Bayesian hierarchical modelling, hidden Markov models, and causality. Examples will be drawn from diverse areas including finance, medicine, forensics, economics and genetics, and students will be introduced to software for graphical modelling (basic knowledge of the software R is a prerequisite).

General Description of the Unit

This unit is held in weeks 7-12 of Teaching Block 1

For complex statistical models it is helpful to lay out all the variables of interest in a diagram.
The resulting graphical model then becomes a useful tool for understanding the relationships between different parts of the model, and helps to suggest techniques for analysis. This unit will study the theory of graphical modelling and apply it to several areas of interest, including:

  • using graphical models to facilitate and accelerate computations: complex highdimensional models arise for example in form of expert systems, where the relationships between factors reflect experts' knowledge, or in the context of Baysian hierarchical modelling. Both types of models are very naturally expressed using graphical models and the graphical structure can be exploited to simplify complex computations.
  • using graphical models for causal reasoning and inference: having a glass of red wine per day is correlated with a reduced risk of heart disease, but is the red wine really causing the reduced risk, or is it simply also a symptom of some other causal factor? In this course causality will also be studied using the language of graphical models.
  • searching for (graphical) structure: in many applications, e.g. genetics, it is the dependence structure among variables or factors itself that is of interest. When represented graphically, the model search can be carried out in a systematic and easy to intepret fashion. Several software packages exist for investigating graphical models. Students on this course will learn to use one of these packages (Hugin, WinBUGS, gR or GRAPPA)
    to perform inference on graphical models.

Relation to Other Units

This unit requires a basic background in probability and statistics, but no prior graph theory is needed. Parts of this unit will refer to Bayesian analysis so that some prior knowledge in this topic will be helpful. An ability and willingness to carry out some basic programming is assumed.

Teaching Methods

Lectures (theory and practical problems) supported by exercise sheets, some of which involve computer practical work with appropriate statistical packages.

Learning Objectives

The students will be able to:

  • Describe the language of graphical modelling.
  • Construct directed acyclic graphs (DAGs) for statistical models.
  • Identify properties of statistical models from the structure of the DAG.
  • Demonstrate the usefulness of graphical models in Bayesian hierarchical models, expert systems, hidden Markov models, causal reasoning and model search.
  • Formulate and fit graphical models using WinBUGS and elements of the gR family of R packages.

Assessment Methods

Assessment will be by means of homework exercises as well as an extended project (to be complete in weeks 13-14), in which graphical modelling is used to investigate one or more practical problems. The student will present a report describing the relevant graphical modelling theory, the model that was used to analyse the data, and the results of the analysis. The assessment criteria for the project will be based on a suitably modified version of the current Mathematics Department Project Assessment form. The written report will count for 90% of the assessment mark and the homework exercises will count for 10%. Both the written report and the exercises will be marked by the member of staff in charge of the unit together with an independent second marker.

Award of Credit Points

Credit points will be awarded if the unit mark is a pass (at least 50).

Transferable Skills

Computing, critical thinking especially regarding causal reasoning, and the ability to give precise mathematical formulations to a variety of problems. Furthermore, writing skills, i.e. the ability to report findings in a coherent report.

Texts

Cowell, Dawid, Lauritzen and Spiegelhalter. Probabilistic Networks and Expert Systems. Springer-Verlag, 1999.
Gelman, Carlin, Stern and Rubin. Bayesian Data Analysis, Chapman & Hall/CRC, 2003.
Gilks, Richardson and Spiegelhalter. Markov Chain Monte Carlo in Practice, Chapman & Hall, 1996.
Lauritzen. Graphical Models, OUP, 1996.
Pearl. Causality: Models, Reasoning and Inference, CUP, 2000.
Whittaker. Graphical Models in Applied Multivariate Statistics, Wiley, 1990.

Syllabus

Introduction to graphs and Markov properties.
Graphical models for computations:
- exact computation with probability propagatio
- approximate computation with McMC
- Bayesian hierarchical models
- hidden Markov models
Graphical models for causal reasoning: causal effect, de-confounding etc.
Learning graphical models (model search): some principles and algorithms.