Bernal S D_2010.pdf - University of Plymouth

XX DEFINITION AND MATHEMATICAL FORMULATION
thus require a prior probability distribution. The prior distributions. P{G) and P{M). indicate
ihat, with no other information available, it is more likely thai Gales are not present (0.8 vs.
0.2); while both stales of the Moon arc equally likely (0.5).
The model assumes when both high-level causes. Gales and Moon, are present {G = l.M —
I), the probability of Waves is higher than when either of the causes is present by itself e.g.
Galeft hut no Moon (G — l,Af = 0). When presented exclusively, Gales is considered to have
a stronger effect over the generation of Waves ihan Moon. All this information is captured
by the conditional probability distribution, in this case a conditional probability table (CPT)
as variables are discrete, over the states of Waves given the stales of Gales and Moon, i.e.
I'{W\G.M).
At the same time. Waves acts as the cause of ihe two lower level effects: the presence of fishing
activity, which is affected negatively by big waves, e.g.fishermen at a pier/beach or small fishing
boats; and the presence of surfing activity, a sport which strongly benefits from big waves. The
presence of fishing activity is denoted by die variable Fishing {F), while the presence of surling
activity is denoted by the variable Surfing {S).
Crucially, the stale of the parent node. Waves, is a determinant factor for the stale of both child
nodes. Fishing and Surfing. The causality dependency between the stale of the node Fishing
with respect to the state of its parent nixle Waves is given by the CPT P{F\W). Analogously,
P{.S\W) represents the conditional probabifity over the states of the node Surfing given the state
of the node Waves.
Using the more formal definition, the Bayesian network in Figure 3.2 can be described as B =
{G,P), where
» G = {V.A) is a directed acyclic graph with a set of vertices V = {G.M.W.F.S};
and a set of arcs A = ((G.W),(M.W),(W,F).(W.S));
• P is the joint probability distribution over V given by,
P{G,M,W,F,S) = P{S\W] • P{F\W) • PiW\G.M) • P{G) • P{W) (3.10)
78