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In this thesis we describe subclasses of a class of graphs with three types of edges, called loopless mixed graphs (LMGs). The class of LMGs contains almost all known classes of graphs used in the literature of graphical Markov models. We focus in particular on the subclass of ribbonless graphs (RGs), which as special cases include undirected graphs, bidirected graphs, and directed acyclic graphs, as well as ancestral graphs and summary graphs. We define a unifying interpretation of independence structure for LMGs and pairwise and global Markov properties for RGs, discuss their maximality, and, in particular, prove the equivalence of pairwise and global Markov properties for graphoids defined over the nodes of RGs. Three subclasses of LMGs (MC, summary, and ancestral graphs) capture the modified independence model after marginalisation over unobserved variables and conditioning on selection variables of variables satisfying independence restrictions represented by a directed acyclic graph (DAG). We derive algorithms to generate these graphs from a given DAG or from a graph of a specific subclass, and we study the relationships between these classes of graphs. Finally, a manual and codes are provided that explain methods and functions in R for implementing and generating various graphs studied in this thesis.

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