Conditional independence and chain event graphs

Graphs provide an excellent framework for interrogating symmetric models of measurement random variables and discovering their implied conditional independence structure. However, it is not unusual for a model to be specified from a description of how a process unfolds (i.e. via its event tree), rather than through relationships between a given set of measurements. Here we introduce a new mixed graphical structure called the chain event graph that is a function of this event tree and a set of elicited equivalence relationships. This graph is more expressive and flexible than either the Bayesian network—equivalent in the symmetric case—or the probability decision graph. Various separation theorems are proved for the chain event graph. These enable implied conditional independencies to be read from the graph's topology. We also show how the topology can be exploited to tease out the interesting conditional independence structure of functions of random variables associated with the underlying event tree.