  Graphical models have proven to be powerful tools for representing
high-dimensional systems of random variables. One example of such a model is
the undirected graph, in which lack of an edge represents conditional
independence between two random variables given the rest. Another example is
the bidirected graph, in which absence of edges encodes pairwise marginal
independence. Both of these classes of graphical models have been extensively
studied, and while they are considered to be dual to one another, except in a
few instances this duality has not been thoroughly investigated. In this paper,
we demonstrate how duality between undirected and bidirected models can be used
to transport results for one class of graphical models to the dual model in a
transparent manner. We proceed to apply this technique to extend previously
existing results as well as to prove new ones, in three important domains.
First, we discuss the pairwise and global Markov properties for undirected and
bidirected models, using the pseudographoid and reverse-pseudographoid rules
which are weaker conditions than the typically used intersection and
composition rules. Second, we investigate these pseudographoid and reverse
pseudographoid rules in the context of probability distributions, using the
concept of duality in the process. Duality allows us to quickly relate them to
the more familiar intersection and composition properties. Third and finally,
we apply the dualization method to understand the implications of faithfulness,
which in turn leads to a more general form of an existing result.
