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#### Graphical models and complex stochastic systems

Lecturer: Peter Green, Statistics group

##### Test
Here it is! | Winbugs code for bonus question | Solutions and comments
##### Syllabus
• Introduction and motivation: complex stochastic systems in science and technology
• Basic ideas of statistical inference. Probability, statistics and data analysis. Models and data. Estimation, confidence intervals and hypothesis testing. Maximum likelihood. Bayesian analysis.
• Conditional independence: axioms, graphical representations, directed acyclic graphs.
• Exchangeability and hierarchical models: Bayesian formulations.
• Bayes nets and expert systems; exact computation by probability propagation.
• Hidden Markov models and state-space models.
• Undirected and chain graph models. Markov random fields and Gibbs distributions. Hammersley-Clifford theorem, spatial models.
• Equilibrium simulation: Markov chain Monte Carlo.
• Model selection and criticism.
• (Learning structure.)
• (Graphical models and causal reasoning.)
• (Role of graphical modelling in computation: specification, visualisation, algorithms.)
Complexity Science graduate programme: course materials page for this module
Related unit: M6002 Graphical modelling (weeks 13-18, in Mathematics)
##### Skeleton notes
Lectures: 1 (pdf/ppt) | 2 | 3-4 | 5 (& extra figs) | 6 (pdf/ppt) | 7 | 8 (pdf/ppt) | 9 | 10
##### Demo code
Brief instructions
Lecture 2: binomial likelihood | normal likelihood | comparing estimators
Lecture 3: likelihood for mixture of two normals | beta-binomial
Lecture 6: 'Asia' expert system | biased coins | hierarchical binomial model | paternity query | mixed trace forensic problem
Lecture 7: 2 state Markov chain | simple hidden Markov model
Lecture 9: 3 Winbugs demos
Exercises 1: random walk | percolation
##### Software
R | gR | Grappa | Hugin (expensive) | Hugin 5.7 (free) | WinBugs
Tutorial on R
To run R (version 2.6.0), Hugin (version 5.7) or WinBugs (version 1.4.3) on the BCCS PC's, look under Start | All Programs | Engineering Apps
##### Exercises
Sheet: 1 | 2 | 3 | 4 | 5
##### Books
• Barndorff-Nielsen, Cox and Klüppelberg (eds.) Complex Stochastic Systems, Chapman and Hall, London, 2001.
• Cappé, Moulines and Rydén. Inference in Hidden Markov Models. Springer, 2005.
• 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.
• Green, Hjort and Richardson. Highly Structured Stochastic Systems, OUP, 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.
• Titterington (ed.) Complex Stochastic Systems and Engineering, IMA Conference Series No. 54. OUP, 1995.
• Whittaker. Graphical Models in Applied Multivariate Statistics, Wiley, 1990.
 Peter Green, Department of Mathematics, University of Bristol, Bristol, BS8 1TW, UK. Email link Telephone: +44 (0)117 928 7967; Fax: +44 (0)117 928 7999