Ms Renee Iannotti

Honours thesis

Graphical models for discrete and continuous multivariate data

Graphical models are a flexible but readily interpreted class of models for multivari- ate data and are valuable tools for high dimensional data sets. Graphical modelling, the set of techniques based on fitting graphical models to data, is a form of statistical modelling which characterises the independence structure of probability models us- ing graphs. Graphical models can accommodate data containing both discrete and continuous variables in a unified mixed variable framework. Decomposable mod- els are a highly important subclass of graphical models particularly for data sets containing a large number of variables. The focus here is primarily on undirected graphical models which are appropriate when the associations between the model variables are assumed to be symmetric. Supported by examples this thesis will dis- cuss conditional independence, the fundamental notion of graphical modelling, and the necessary graph theoretic terminology before developing in detail the theory of graphical models, reworking and elaborating previous work in the area. Graphical models can be interpreted in terms of conditional independence relations which can be read directly off the independence graph corresponding to a given graphical model. Decomposable models are valuable from a practical perspective as they break high dimensional problems into a series of lower dimensional problems. The joint density functions for these models factorise into the product of marginal density functions, their joint likelihoods can be factorised and they admit closed form maximum likelihood estimates.