Abstract:

Graphs, that consist of nodes for random variables and of edges that
couple node pairs,  may capture different types of independence structure,
depending on the type and number of of edges present. Given the set of
independence statements defining such  graphs, all other independence
statements belonging to the structure are obtainable via the  global Markov
property of the graph.

There are several types of more complex questions concerning
independence structures that arise in particular  when the results of
different studies are to be compared. One example occurs when
single response variables are changed into joint responses, another
when we need  to decide how the independence structure changes for a
selected subset of variables  and after  fixing levels  of some other variables.

We  use operators for binary matrix representations of graphs by which
we transform graphs repeatedly. Thereby, we obtain what we call summary
graphs. They are closed under marginalizing and conditioning and preserve
the independence structure implied by the generating directed
acyclic graph.