Stratified Staged Trees: Modelling, Software and Applications

The thesis is focused on Probabilistic Graphical Models (PGMs), which are a rich framework for encoding probability distributions over complex domains. In particular, joint multivariate distributions over large numbers of random variables that interact with each other can be investigated through PGMs and conditional independence statements can be succinctly represented with graphical representations. These representations sit at the intersection of statistics and computer science, relying on concepts mainly from probability theory, graph algorithms and machine learning. They are applied in a wide variety of fields, such as medical diagnosis, image understanding, speech recognition, natural language processing, and many more. Over the years theory and methodology have developed and been extended in a multitude of directions. In particular, in this thesis different aspects of new classes of PGMs called Staged Trees and Chain Event Graphs (CEGs) are studied. In some sense, Staged Trees are a generalization of Bayesian Networks (BNs). Indeed, BNs provide a transparent graphical tool to define a complex process in terms of conditional independent structures. Despite their strengths in allowing for the reduction in the dimensionality of joint probability distributions of the statistical model and in providing a transparent framework for causal inference, BNs are not optimal GMs in all situations. The biggest problems with their usage mainly occur when the event space is not a simple product of the sample spaces of the random variables of interest, and when conditional independence statements are true only under certain values of variables. This happens when there are context-specific conditional independence structures. Some extensions to the BN framework have been proposed to handle these issues: context-specific BNs, Bayesian Multinets, or Similarity Networks citep{geiger1996knowledge}. These adopt a hypothesis variable to encode the context-specific statements over a particular set of random variables. For each value taken by the hypothesis variable the graphical modeller has to construct a particular BN model called local network. The collection of these local networks constitute a Bayesian Multinet, Probabilistic Decision Graphs, among others. It has been showed that Chain Event Graph (CEG) models encompass all discrete BN models and its discrete variants described above as a special subclass and they are also richer than Probabilistic Decision Graphs whose semantics is actually somewhat distinct. Unlike most of its competitors, CEGs can capture all (also context-specific) conditional independences in a unique graph, obtained by a coalescence over the vertices of an appropriately constructed probability tree, called Staged Tree. CEGs have been developed for categorical variables and have been used for cohort studies, causal analysis and case-control studies. The user’s toolbox to efficiently and effectively perform uncertainty reasoning with CEGs further includes methods for inference and probability propagation, the exploration of equivalence classes and robustness studies. The main contributions of this thesis to the literature on Staged Trees are related to Stratified Staged Trees with a keen eye of application. Few observations are made on non-Stratified Staged Trees in the last part of the thesis. A core output of the thesis is an R software package which efficiently implements a host of functions for learning and estimating Staged Trees from data, relying on likelihood principles. Also structural learning algorithms based on distance or divergence between pair of categorical probability distributions and based on the clusterization of probability distributions in a fixed number of stages for each stratum of the tree are developed. Also a new class of Directed Acyclic Graph has been introduced, named Asymmetric-labeled DAG (ALDAG), which gives a BN representation of a given Staged Tree. The ALDAG is a minimal DAG such that the statistical model embedded in the Staged Tree is contained in the one associated to the ALDAG. This is possible thanks to the use of colored edges, so that each color indicates a different type of conditional dependence: total, context-specific, partial or local. Staged Trees are also adopted in this thesis as a statistical tool for classification purpose. Staged Tree Classifiers are introduced, which exhibit comparable predictive results based on accuracy with respect to algorithms from state of the art of machine learning such as neural networks and random forests. At last, algorithms to obtain an ordering of variables for the construction of the Staged Tree are designed.