Exploiting Contextual Independence In Probabilistic Inference -- Parent Skeletons

Parent Skeletons

Parent Skeletons

Although the definition of a contextual belief network specifies the contextual independence we want, it doesn't give us a way to organize the parent contexts (in much the same way as a belief network doesn't specify the representation of a conditional probability table). We use the concept of a parent skeleton as a way to organize the parent contexts; we want to use the indexing provided by tables while still allowing for the ability to express context-specific independence.

The notion of a parent context is more fine-grained than that of a parent (the set of parents corresponds to many parent contexts). When there is no context-specific independence, we would like to not have to consider the parent contexts explicitly, but consider just the parents. We will use a parent skeleton to cover both parents and parent contexts as special cases, and to interpolate between them, when the independence depends on some context as well as all values of some other variables.

Definition. A parent skeletal pair for variable X is a pair <c,V> where c is a context on the predecessors of X and V is a set of predecessors of X such that X is contextually independent of its predecessors given V and context c. Note that a parent context is c&V=v. A parent skeleton for variable X is a set of parent skeletal pairs, {<c_j,V_j>:0<j <= k}, where the c_j are mutually exclusive and exhaustive (i.e., c_i and c_j are incompatible if i != j, and &_j=1^k c_j == true).

Example. A parent skeleton for E from Example * is { <a,{B}>, <~a&c,{}>, <~a&~c&d,{B}>, <~a&~c&~d,{}>.

Parent skeletons form the basis of a representation for contextual belief networks. For each variable, X, you select a parent skeleton such that for each parent skeleton pair <c_j,V_j> in the parent context, c_j&V_j=v_j is a parent context for X. For each such parent context pair we specify a probability distribution P(X|c_j&V_j=v_j).

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

Parent Skeletons