Unit information: Bayesian Modelling in 2017/18

Unit name	Bayesian Modelling
Unit code	MATH30015
Credit points	20
Level of study	H/6
Teaching block(s)	Teaching Block 2 (weeks 13 - 24)
Unit director	Dr. Gerber
Open unit status	Not open
Pre-requisites	Statistics 2 (MATH20800), Probability 2 (MATH20008)
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

Bayesian statistics is an area that has grown rapidly in popularity over the past 20 years or so largely as a result of computational advances which have made the approach far more applicable. In this unit we will discuss the Bayesian approach to statistical analysis and modelling. We introduce the basic elements of Bayesian theory, beginning with Bayes theorem, and go on to discuss the applications of this approach to statistical modelling. Topics discussed will include the construction of prior and posterior distributions and hierarchical models, large sample inference and connections to non-Bayesian methods, model checking, and a brief introduction to the computational tools which make analysis possible (in particular Markov chain Monte Carlo methods).

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 slightly more complex situations. 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 but not identical.

We will study how to formulate and use such models (answer - by Bayes's theorem!), and then how to actually do that in practice, since we will no longer have conjugacy to help us, as in the first part of this unit. This leads to discussion of Markov chain Monte Carlo (MCMC) techniques, which are powerful and elegant algorithms based on simple ideas of conditional probability. Hierarchical models and MCMC come together in the JAGS software, which can be run within R. We will use JAGS to analyse several famous problems, and also some new ones.

Intended Learning Outcomes

The students will be able to:

• Understand and explain the theoretical basis for and range of applications of the Bayesian approach to statistical modelling;
• Describe and construct realistic and appropriate statistical models to describe a wide variety of modelling situations;
• Use and understand appropriate computational methodology within a Bayesian framework.
• 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

Teaching Information

Lectures (theory and practical problems) supported by handouts and worksheets, some of which involve computer practical work with R and JAGS. A weekly Office Hour. Regular formative problem sheets.

Assessment Information

20% computing assessment, 80% examination (2.5 hours)

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

Reading and References

Recommended:

1. Gelman, A., Carlin, J.B., Stern, H.S. and Rubin, D.B. Bayesian Data Analysis, Chapman and Hall.
2. Robert, C.P. The Bayesian Choice, Springer-Verlag.

Further:

1. J. M. Bernardo and A. Smith. Bayesian Theory, Wiley.
2. J.-M. Marin and C. P. Robert. Bayesian Core: A Practical Approach to Computational Bayesian Statistics, Springer-Verlag.
3. Robert, C.P. and Casella, G., Monte Carlo Statistical Methods, Springer-Verlag.
4. D. Gamerman. Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Chapman and Hall.
5. Gilks, W.R., Richardson, S. and Spiegelhalter, D. Markov Chain Monte Carlo in Practice, Chapman and Hall.
6. Morgan, B.J.T. Elements of Simulation, Chapman and Hall.