340 14 Bayesian Networks
The techniques above are concerned with the specification of PDs. A CPD
is a function from the possible values of the parent nodes to PDs on the node.
If there are only a few possible values of the parent nodes (as in the diagnostic
example in figure 14.1), then explicitly listing all of the PDs is feasible. Many
BN tools have no other mechanism for specifying CPDs. When the number
of possible values of the parent nodes is large or even infinite, then the CPD
may be much better specified using a function. In the infinite case, one has
no choice but to use this technique. Curve-fitting techniques such as least-
squares analysis can be used to choose the function based on the available
data.
A BN with both discrete and continuous nodes is called ahybrid BN.The
diagnostic BN example above is a hybrid BN. When continuous nodes are
dependent on discrete nodes, inference will produce a compound (mixed)
Gaussian distribution. Such a distribution is the result of a compound pro-
cess in which one of a finite set of Gaussians is selected according to a PD, and
then a value is chosen based on the particular Gaussian that was selected.
If a discrete node is dependent on continuous nodes, then the discrete node
can be regarded as defining aclassifiersince it takes continuous inputs and
produces a discrete output whichclassifiesthe inputs. The CPDs for this situ-
ation are usually chosen to be logistic/softmax distributions. Connectionist
networks (also called neural networks) are an example of this.
BNs are not the only graphical representation for stochastic models. Undi-
rected graphical models, also called Markov random fields (MRFs) or Mar-
kov networks, are also used, especially in the physics and vision communi-
ties.
One application of BNs is to assist in decision making. To make a decision
based on evidence one must quantify the risk associated with the various
choices. This is done by using a utility function. It is possible to model
some utility functions by addingvalue nodes(also calledutility nodes)toa
BN and linking them with dependency edges to ordinary BN nodes and to
other utility nodes. The result of a decision is an action that is performed,
and these can also be represented graphically by addingdecision nodesand
edges to a BN augmented with utility nodes. A BN augmented with utility
and action nodes is called aninfluence diagram(also called arelevance diagram)
(Howard and Matheson 1981). An influence diagram can, in principle, be
used to determine the optimal actions to perform so as to maximize expected
utility.