The dynamic chain event graph.

The Chain Event Graph (CEG) is a type of tree-based graphical model that accommodates all discrete Bayesian Networks as a particular subclass. It has already been successfully used to capture context-specific conditional independence structures of highly asymmetric processes in a way easily appreciated by domain experts. Being built from a tree, a CEG has a huge number of free parameters that makes the class extremely expressive but also very large. Exploring the enormous CEG model space then makes it necessary to design bespoke algorithms for this purpose. All Bayesian algorithms for CEG model selection in the literature are based on the Dirichlet characterisation of a family of CEGs spanned by a single event tree. Here I generalise this framework for a CEG model space spanned by a collection of different event trees. A new concept called hyper-stage is also introduced and provides us with a framework to design more efficient algorithms. These improvements are nevertheless insufficient to scale up the model search for more challenging applications. In other contexts, recent analyses of Bayes Factor model selection using conjugate priors have suggested that the use of such prior settings tends to choose models that are not sufficiently parsimonious. To sidestep this phenomenon, non-local priors (NLPs) have been successfully developed. These priors enable the fast identification of the simpler model when it really does drive the data generation process. In this thesis, I define three new families of NLPs designed to be applied specifically to discrete processes defined through trees. In doing this, I develop a framework for a CEG model search which appears to be both robust and computationally efficient. Finally, I define a Dynamic Chain Event Graph (DCEG). I develop object-recursive methods to fully analyse a particularly useful and feasibly implementable new subclass of these models called the N Time-Slice DCEG (NT-DCEG). By exploiting its close links with the Dynamic Bayesian Network I show how the NT-DCEG can be used to depict various structural and Granger causal hypotheses about a studied process. I also show how to construct from the topology of this graph intrinsic random variables which exhibit context-specific independences that can then be checked by domain experts. Throughout the thesis my methods are illustrated using examples of multivariate processes describing inmate radicalisation in a prison, and survey data concerning childhood hospitalisation and booking a tourist train