Bernal S D_2010.pdf - University of Plymouth

3 J. DEFINITION AND MATHEMATICAL FORMULATION
its descendants is available. This rule tells us that if nothing is known about a common
effect of two (or more) causes, then the cau.ses are independent. In other words Gales is
not an indicator of Moon, and vice versa. However, as soon as some evidence is available
on a common effect, the causes become dependent. If, for example, we receive some
information on the state of VVijvcv. then Gales and Moon become compeiing explanations
for this effect. Thus, receiving infonnation about one of the causes either confirm.s or
dismisses the other one as the cause of Waves. Note that even if the initial information
about the Waves is not reliable (soft evidence), Gales and Moon still become dependent.
The property of converging connections, where information about the state of a parent
node provides an explanation for an observed effect, and hence contirms or dismisses
another parent node as the cause of the effect, is often referred to as the explaining away
effect or as inlercausal inference. I-or example, knowing Gales are present strongly sug­
gests these are responsible for the Waves, hence explaining away the Moon as die cause
of the Waves.
Critically, one of the fundamental properties of directed acyclic graphs (Bayesian networks) is
that for a given node X. the set of its parents, rijf, d-separales this node from all other subsets
Y with no descendants of X, such thatX ±L f | Ilx. In other words, each node in the network is
conditionally independent from its non-descendants, given its parents. This allows us to obtain
the factorization of the joint probabiHly distribution shown in h^^uation (3.9), as the following
property is satisfied;
3.3.2.3 Cycles and acyclic graphs
p{x,\nx„Y) = p{x^nx^ (3.11)
A chain consists of a series of nixies where each successive node in the chain is connected to
the previous one by an edge. A path is a chain where each connection edge in the chain has the
same directionality, i.e. all are serial connections. For example, nodes M —> W -^ 5 (l-igure 3.2
form a path; while nodes M ^W