  It is well-known that the notion of (strong) conditional independence (CI) is
too restrictive to capture independencies that only hold in certain contexts.
This kind of contextual independency, called context-strong independence (CSI),
can be used to facilitate the acquisition, representation, and inference of
probabilistic knowledge. In this paper, we suggest the use of contextual weak
independence (CWI) in Bayesian networks. It should be emphasized that the
notion of CWI is a more general form of contextual independence than CSI.
Furthermore, if the contextual strong independence holds for all contexts, then
the notion of CSI becomes strong CI. On the other hand, if the weak contextual
independence holds for all contexts, then the notion of CWI becomes weak
independence (WI) nwhich is a more general noncontextual independency than
strong CI. More importantly, complete axiomatizations are studied for both the
class of WI and the class of CI and WI together. Finally, the interesting
property of WI being a necessary and sufficient condition for ensuring
consistency in granular probabilistic networks is shown.
