Contextual Independence

Contextual Independence

In this section we give a formalization of contextual independence. This notion was first introduced into the influence diagram literature [29]. We base our definitions on the work of [5].

Definition. Given a set of variables C, a context on C is an assignment of one value to each variable in C. Usually C is left implicit, and we simply talk about a context. We would say that C are the variables of the context. Two contexts are incompatible if there exists a variable that is assigned different values in the contexts; otherwise they are compatible. We write the empty context as true.

Definition. [5] Suppose X, Y, Z and C are sets of variables. X and Y are contextually independent given Z and context C=c, where c in dom(C), if

P(X|Y=y₁&Z=z₁&C=c) = P(X|Y=y₂&Z=z₁&C=c)

for all y₁,y₂ in dom(Y) for all z₁ in dom(Z) such that P(Y=y₁&Z=z₁ &C=c)>0 and P(Y=y₂&Z=z₁ &C=c)>0.
We also say that X is contextually independent of Y given Z and context C=c. Often we will refer to the simpler case when the set of variables Z is empty; in this case we say that X and Y are contextually independent given context C=c.

Example. Given the belief network and conditional probability table of Figure *,

E is contextually independent of {C,D,Y,Z} given context a&b.
E is contextually independent of {C,D,Y,Z} given {B} and context a.
E is not contextually independent of {C,D,Y,Z} given {A,B} and the empty context true.
E is contextually independent of {B,D,Y,Z} given context ~a&c.
E is contextually independent of {A,B,C,D,Y,Z} given B and context ~a&~c &d.

David Poole and Nevin Lianwen Zhang,Exploiting Contextual Independence In Probabilistic Inference, Journal of Artificial Intelligence Research, 18, 2003, 263-313.

Contextual Independence