On Probabilistic Independence Models and Graphs

The main purpose of this talk is to explore the relationship between the set of conditional independence statements induced by a probability distribution and the set of separations induced by graphs as studied in graphical models. I define one general type of graph and one separation criterion, and show that almost all known types of graphs and separation criteria are a special case of these. I introduce the concepts of Markov property and faithfulness, and provide conditions under which a given probability distribution is Markov or faithful to a graph. I discuss the implications of these conditions in different areas of statistics, probability theory, and machine learning.