The Discrete Case: Multinomial Bayesian Networks 21
depend directly on each other, there will be a single arc connecting the nodes
corresponding to those two variables. If the dependence is indirect, there will
be two or more arcs passing through the nodes that mediate the association.
In general, two sets X and Y of variables are independent given a third set Z
of variables if there is no set of arcs connecting them that is not blocked by the
conditioning variables. Conditioning on Z is equivalent to fixing the values of
its e lements, so that they are known quantities. In other words, the X and
Y are separated by Z, which we denote with X ⊥⊥
G
Y | Z. Given that BNs
are based on DAGs, we speak of d-separation (directed separation); a formal
treatmen