Extending Graphically Specified Models

In Chapter 6 we introduced graphically specified models using undirected or directed graphs to describe the dependency structures of probabilistic models. We have seen that not every probabilistic model can be specified by a perfect map. Thus, in general, graphs can only be thought of as minimal independence maps (I-maps), from which every conditional independence statement (CIS) derived from the graph holds in the associated probability model, though some CISs in the probability model may not be represented by the graph. Consequently, the main limitation of graphical models is that they can only represent certain types of independence structures. The following example illustrates this limitation.