Graphical models and complex stochastic systems

Lecturer: Peter Green, Statistics group

Test

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.)

Skeleton notes

Demo code

Software

Exercises

Some relevant work of mine

• Further hidden Markov model cryptanalysis - extending forwards/backwards recursions to HMMs where the data sequence and hidden state sequence are not perfectly aligned, with a cryptanalysis application.
• Hidden Markov models and disease mapping - using a spatial version of a HMM in an application to spatial epidemiology
• Inference from genome-wide association studies using a novel Markov model - a structured statistical model for a problem in genetic epidemiology, using a HMM as part of the model
• Reversible jump Markov chain Monte Carlo computation and Bayesian model determination - MCMC methodology for variable dimension problems, e.g. in model determination
• Enumerating the junction trees of a decomposable graph - as we said in the course, the junction tree is not unique - but just how many different junction trees are there for a given (decomposable) graph?

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