Project IV: Graphical Models for Multivariate Statistics - Jonathan Cumming
Description
Graphical modelling is a form of multivariate analysis that uses graphs to represent statistical models. Realistic multivariate problems in statistics are characterised by the presence of multiple variables, and a complex web of relationships between them. A graphical model encodes such problems by representing the variables as the nodes of the graph, and the relationships between pairs of variables as edges representing statements of conditional dependence. Underlying this graph structure is then a joint multivariate distribution, which provides the basis for statistical inference.
Graphical models come in a variety of flavours depending on whether we assign a direction to the edges of the graph. Directed graphs lend themselves to a Bayesian treatment and so are often known as Bayesian networks, whereas undirected graphical models, also known as Markov random fields, often follow a frequentist approach based on likelihood and parameter estimation. Continuous and discrete-valued data must also be handled differently using methods based on the multivariate Normal distribution and log-linear models respectively.
A simplified Bayesian network model for the diagnosis of scurvy and
plague.
Graphical modelling is a versatile technique which has been used in a wide range of application domains, including: decision support, medical and fault diagnosis, web search, image analysis, speech recognition, natural language processing, decoding of messages sent over a noisy communication channel, and robot navigation.
In this project, you will begin by studying how multivariate statistical problems can be expressed as graphical models, learn about Graphical Gaussian models for multivariate Normally-distributed data, and apply this knowledge to the analysis of real data sets. There are then several further topics which you could pursue depending your interest. Some examples would be:
- Graphical modelling methods for discrete- and mixed-valued data
- Directed graphical models and Bayesian networks
- Methods for learning the structure of a graphical model from data
- Methods for parameter estimation, and fitting a given graphical structure to data
- Techniques for updating such models, e.g. in the presence of new data, and making inferences
- Analysis and interpretation of a substantial data set
This project has a focus on statistical methodology and data analysis. Familiarity with the statistical package R, general statistical concepts, and data analysis are essential.
Prerequisites
Statistical Concepts II, Statistical Methods III
Would be nice/might help, but not essential
Bayesian Statistics III/IV
Web Resources
- The wikipedia pages on Graphical models, Bayesian networks, and Markov random fields are a good starting point
- The R task view on graphical modelling gives a comprehensive list of relevant R packages.
Books
Many books contain suitable introductory material:
- Probabilistic Graphical Models by Koller and Friedman gives a textbook treatment of the subject.
- Graphical Models with R focuses on the practical details of using these methods in R. An online tutorial version of this book gives a brief practical overview of the same material.
- The library has several books on this topic. See here.
- Graphical models by Lauritzen delves rather deeper into the theory. Not for the faint-hearted!