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A Property of Independency Relations Induced by Probabilistic Distributions with Binary Variables

Paz A.

Fundamenta Informaticae

2006

Vol. 73, nr 1-2

229-236

The relationship between graphoid independency relations (defined in the text) and such relations induced by Probabilistic Distributions (PD) with binary random variables is investigated. It is shown that there are axioms that are sound for a subset of PD-induced relations with binary variables and are independent of the Graphoid axioms (cannot be logically derived from those axioms).

Representation of irrelevance relations by annotated graphs

Paz A., Geva R.Y., Studeny M.

Fundamenta Informaticae

2000

Vol. 42, Nr 2

149-199

Irrelevance relations are sets of statements of the form: given that the `value' of Z is known, the `values' of Y can add no further information about the `values' of X. Undirected Graphs (UGs), Directed Acyclic Graphs (DAGs) and Chain Graphs (CGs) were used and investigated as schemes for the purpose of representing irrelevance relations. It is known that, although all three schemes can approximate irrelevance, they are inadequate in the sense that there are relations which cannot be fully represented by anyone of them. In this paper annotated graphs are defined and suggested as a new model for graphical representation. It is shown that this new model is a proper generalization of the former models: any irrelevance relation that can be represented by either one of the previous models can also be represented by an annotated graph, and there are relations that can be represented by an annotated graph but cannot be represented by either one of the former models. The question of whether this new model is powerful enough to represent all the irrelevance relations, as well as some other related questions, is still open.