We use the following conventions to identify counts in the database and a network structure . Let () be the cardinality of . We use to denote the cardinality of the parent set of in , that is, the number of different values to which the parents of can be instantiated. So, can be calculated as the product of cardinalities of nodes in , . Note implies . We use (, ) to denote the number of records in for which takes its th value.We use (, , ) to denote the number of records in for which takes its th value and for which takes its th value. So, . We use to denote the number of records in .
Let the entropy metric of a network structure and database
be defined as
AIC metric The AIC metric
of a Bayesian network
structure for a database is
MDL metric The minimum description length metric of a Bayesian network structure for a database is is defined as
Bayesian metric
The Bayesian metric of a Bayesian network structure for a database is