Ms Renee Iannotti
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.