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Axioms of Causal Relevance
 Artificial Intelligence
, 1996
"... This paper develops axioms and formal semantics for statements of the form "X is causally irrelevant to Y in context Z," which we interpret to mean "Changing X will not affect Y if we hold Z constant." The axiomization of causal irrelevance is contrasted with the axiomization ..."
Abstract

Cited by 52 (13 self)
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This paper develops axioms and formal semantics for statements of the form "X is causally irrelevant to Y in context Z," which we interpret to mean "Changing X will not affect Y if we hold Z constant." The axiomization of causal irrelevance is contrasted with the axiomization of informational irrelevance, as in "Learning X will not alter our belief in Y , once we know Z." Two versions of causal irrelevance are analyzed, probabilistic and deterministic. We show that, unless stability is assumed, the probabilistic definition yields a very loose structure, that is governed by just two trivial axioms. Under the stability assumption, probabilistic causal irrelevance is isomorphic to path interception in cyclic graphs. Under the deterministic definition, causal irrelevance complies with all of the axioms of path interception in cyclic graphs, with the exception of transitivity. We compare our formalism to that of [Lewis, 1973], and offer a graphical method of proving theorems abou...
Axiomatic Characterization of Directed Graphs
, 1994
"... While graphs are normally defined in terms of the 2place relation of adjacency, we take the 3place relation of interception as the basic primitive of their definition. building on results previously obtained for undirected graphs, this paper establishes an axiomatic characterization of 3place rel ..."
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Cited by 2 (1 self)
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While graphs are normally defined in terms of the 2place relation of adjacency, we take the 3place relation of interception as the basic primitive of their definition. building on results previously obtained for undirected graphs, this paper establishes an axiomatic characterization of 3place relations that lend themselves to representation in terms of path interception in directed graphs, thus providing a new characterization of graphs. ffl We will refer to our previous characterization paper from June `93 as [PPU] ffl Given a directed graph (not necessarily acyclic) define RG (X; Z; Y ) as "all the directed paths from X to Y are intercepted by Z ." 1 Axioms X:2:1 R(X;ZY;W ) ! R(X;Z;Y ) R(X;Z;W ) X:2:2 R(XW;Z;Y ) ! R(X;Z;Y ) R(W;Z;Y ) X:3 R(X;Z;Y ) ! R(X;ZW;Y ) all W X:4:1 R(X;ZW;Y ) R(X;ZY;W ) ! R(X;Z;WY ) X:4:2 R(X;ZW;Y ) R(W;XZ;Y ) ! R(XW;Z;Y ) X:5 R(X;Z;Y ) ! R(a; Z; Y ) R(X;Z; a) all a 62 X [ Z [ Y X:2 = Decomposition; X:3 = Strong union X:4 = Intersection; X:5 = Tran...
A Universal Algebraic Approach for Conditional Indepencence
 ANNALS OF THE INSTITUTE OF STATISTICAL MATHEMATICS
"... In this paper we show that elementary properties of joint probability density functions naturally induce a universal algebraic structure suitable for studying probabilistic conditional independence (PCI) relations. We call this algebraic system the cain. In the cain algebra, PCI relations are repre ..."
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In this paper we show that elementary properties of joint probability density functions naturally induce a universal algebraic structure suitable for studying probabilistic conditional independence (PCI) relations. We call this algebraic system the cain. In the cain algebra, PCI relations are represented in equational forms. In particular we show that the cain satisfies the axioms of the graphoid of Pearl and Paz