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Graphoid
A graphoid is a set of statements of the form, "X is irrelevant to Y given that we know Z" where X, Y and Z are sets of variables. The notion of "irrelevance" and "given that we know" may obtain different interpretations, including probabilistic, relational and correlational, depending on the application. These interpretations share common properties that can be captured by paths in graphs (hence the name "graphoid"). The theory of graphoids characterizes these properties in a finite set of axioms that are common to informational irrelevance and its graphical representations.
Judea Pearl and Azaria Paz coined the term "graphoids" after discovering that a set of axioms that govern conditional independence in probability theory is shared by undirected graphs. Variables are represented as nodes in a graph in such a way that variable sets X and Y are independent conditioned on Z in the distribution whenever node set Z separates X from Y in the graph. Axioms for conditional independence in probability were derived earlier by A. Philip Dawid and Wolfgang Spohn. The correspondence between dependence and graphs was later extended to directed acyclic graphs (DAGs) and to other models of dependency.
A dependency model M is a subset of triplets (X,Z,Y) for which the predicate I(X,Z,Y): X is independent of Y given Z, is true. A graphoid is defined as a dependency model that is closed under the following five axioms:
A semi-graphoid is a dependency model closed under 1–4. These five axioms together are known as the graphoid axioms. Intuitively, the weak union and contraction properties mean that irrelevant information should not alter the relevance status of other propositions in the system; what was relevant remains relevant and what was irrelevant remains irrelevant.
Conditional independence, defined as
is a semi-graphoid which becomes a full graphoid when P is strictly positive.
A dependency model is a correlational graphoid if in some probability function we have,
where is the partial correlation between x and y given set Z.
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Graphoid
A graphoid is a set of statements of the form, "X is irrelevant to Y given that we know Z" where X, Y and Z are sets of variables. The notion of "irrelevance" and "given that we know" may obtain different interpretations, including probabilistic, relational and correlational, depending on the application. These interpretations share common properties that can be captured by paths in graphs (hence the name "graphoid"). The theory of graphoids characterizes these properties in a finite set of axioms that are common to informational irrelevance and its graphical representations.
Judea Pearl and Azaria Paz coined the term "graphoids" after discovering that a set of axioms that govern conditional independence in probability theory is shared by undirected graphs. Variables are represented as nodes in a graph in such a way that variable sets X and Y are independent conditioned on Z in the distribution whenever node set Z separates X from Y in the graph. Axioms for conditional independence in probability were derived earlier by A. Philip Dawid and Wolfgang Spohn. The correspondence between dependence and graphs was later extended to directed acyclic graphs (DAGs) and to other models of dependency.
A dependency model M is a subset of triplets (X,Z,Y) for which the predicate I(X,Z,Y): X is independent of Y given Z, is true. A graphoid is defined as a dependency model that is closed under the following five axioms:
A semi-graphoid is a dependency model closed under 1–4. These five axioms together are known as the graphoid axioms. Intuitively, the weak union and contraction properties mean that irrelevant information should not alter the relevance status of other propositions in the system; what was relevant remains relevant and what was irrelevant remains irrelevant.
Conditional independence, defined as
is a semi-graphoid which becomes a full graphoid when P is strictly positive.
A dependency model is a correlational graphoid if in some probability function we have,
where is the partial correlation between x and y given set Z.