4 Conclusions and Further Research

9] S.L. Lauritzen and D.J. Spiegelhalter. Local computations with probabilities on graphical structures and their applications to expert systems (with discussion). 13 algorithms currently known for inference in probabilistic networks. IDAGs can also be helpful for constructing a DAG from a given independency model. For the construction of a DAG from an independency model, in 1] an algorithm has been presented that optimizes a total ordering of a set of variables by applying simple operations to. Changing such an ordering implies a change in the causal input list generated from the ordering, and thus a change in the associated DAG G and the represented in-dependency model M. The strategy is to apply an operation to only if the represented independency model after the operation contains the one before the operation. This strategy has been shown to yield a DAG with a minimal number of arcs in a local minimum. In order to nd the global minimum, we believe the IDAG formalism can be of help, since after an operation on the total< ordering , independency inducing arcs can be added in order to let the new independency model contain the old one. Conditional independence relations are used to reeect irrelevancy in knowledge based systems. The notion of irrelevancy that is reeected by conditional independence is global in the sense that it applies to all values of variables. However, variables may be independent for certain combinations of values, yet dependent for other combinations of values. It may be worth while to look for formalisms and eecient inference and explanation algorithms that explore these independencies; introduction of a special kind of arc similar to the independency inducing arc employed by the IDAG formalism may be used to reeect such independencies. Acknowledgement I thank Linda van der Gaag for her many helpfull remarks that improved the presentation of this paper considerably. References 1] R.R. Bouckaert. Optimizing causal orderings for generating DAGs from data. 12 y. We know that G is an I-map of M, 11]. Adding arcs, thus obtaining the embedded DAG of G 0 , still results in an I-map. So, G 0 is an I-map of M by Lemma 3.2. Observe that no dependencies are added by removing the arc y ! x and that G was supposed to be a P-map. Therefore, G 0 also has to be a P-map. The validity of the independency inducing arc-removal rule follows from …