## 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.